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Limit of a Function and
One-sided limits
Mathematics 53
Institute of Mathematics (UP Diliman)
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
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Introduction
Given a function f(x) and a
R ,
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Introduction
Given a function f(x) and a
R ,
what is the value of f at x near a,
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Introduction
Given a function f(x) and a
R ,
what is the value of f at x near a,
but not equal to a?
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Illustration 1
Consider f(x) = 3x 1.
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 1
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 5
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
Based on the table, as x gets closer and closer to 1,
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Illustration 1
Consider f(x) = 3x 1.What can we say about values of f(x) for values of x near 1 but not equal to 1?
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
x f(x)
2 51.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
Based on the table, as x gets closer and closer to 1, f(x) gets closer and closer
to 2.
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Ill i 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill t ti 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Ill stration 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
0 10.5 0.5
0.9 1.7
0.99 1.97
0.99999 1.99997
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
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Illustration 1
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Illustration 1
x f(x)
2 5
1.5 3.5
1.1 2.3
1.001 2.003
1.00001 2.00003
1 1 2 3
1
1
2
3
4
As x gets closer and closer to 1, f(x) gets closer and closer to 2.
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Illustration 2
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Illustration 2
Consider: g(x) =3x2 4x + 1
x 1
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Illustration 2
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Illustration 2
Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
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Illustration 2
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Illustration 2
Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
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Illustration 2
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Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
1 1 2 31
1
2
3
4
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Illustration 2
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Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 2
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Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 2
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Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 2
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Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 2
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Consider: g(x) =3x2 4x + 1
x 1=
(3x 1)(x 1)x 1
= 3x 1, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1, g(x) gets closer and closer to 2.
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Illustration 3
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Consider: h(x) =
3x 1, x = 10, x = 1
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Illustration 3
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Consider: h(x) =
3x 1, x = 10, x = 1
1 1 2 31
1
2
3
4
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Illustration 3
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Consider: h(x) =
3x 1, x = 10, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x = 10, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x = 10, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1,
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Illustration 3
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Consider: h(x) =
3x 1, x = 10, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1, h(x) gets closer and closer to 2.
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Limit
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Limit
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Intuitive Notion of a Limit
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Limit
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Intuitive Notion of a Limita R , L R
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Limit
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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a
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Limit
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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a
The limit of f(x) as x approaches a is L
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Limit
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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a
The limit off(x)
asx
approachesa
isL
if the values of f(x) get closer and closer to L as x assumes values getting closer
and closer to a but not reaching a.
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Limit
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Intuitive Notion of a Limita R , L Rf(x): function defined on some open interval containing a, except possibly at a
The limit off(x)
asx
approachesa
isL
if the values of f(x) get closer and closer to L as x assumes values getting closer
and closer to a but not reaching a.
Notation:limxa f(x) = L
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4limx1
(3x 1)
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4limx1
(3x 1) = 2
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4limx1
(3x 1) = 2
Note: In this case, limx1
f(x)
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Examples
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f(x) = 3x 1
1 1 2 3
1
1
2
3
4limx1
(3x 1) = 2
Note: In this case, limx1
f(x) = f(1).
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Examples
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g(x) =3
x
2
4
x +1
x 1
1 1 2 31
1
2
3
4
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Examples
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g(x) =3
x
2
4
x+ 1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x + 1x 1
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Examples
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g(x) =3x2
4x + 1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x + 1x 1 = 2
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Examples
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g(x) =3x2
4x + 1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x + 1x 1 = 2
Note: Though g(1) is undefined,limx1
g(x) exists.
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Examples
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h(x) =
3x
1, x
= 1
0, x = 1
1 1 2 31
1
2
3
4
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Examples
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h(x) =
3x
1, x
= 1
0, x = 1
1 1 2 31
1
2
3
4
limx1
h(x)
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Examples
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h(x) =
3x
1, x
= 1
0, x = 1
1 1 2 31
1
2
3
4
limx1
h(x) = 2
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Examples
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h(x) =
3x
1, x
= 1
0, x = 1
1 1 2 31
1
2
3
4
limx1
h(x) = 2
Note: h(1) = limx1 h(x).
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Some Remarks
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Remark
In finding limxa f(x):
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close to a but not exactly at a.
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close to a but not exactly at a.
Thus, limxa f(x) is NOT NECESSARI LY the same as f(a).
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Some Remarks
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Remark
In finding limxa f(x):
We only need to consider values of x very close to a but not exactly at a.
Thus, limxa f(x) is NOT NECESSARI LY the same as f(a).
We let x approach a from BOTH SIDES of a.
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Iff(x)
does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
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Some Remarks
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Iff(x)
does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
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Some Remarks
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If f(x) does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40
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Some Remarks
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If f(x) does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40
Some Remarks
7/29/2019 Lecture 2 - Limits and One-Sided Limits
79/290
If f(x) does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1?
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40
Some Remarks
7/29/2019 Lecture 2 - Limits and One-Sided Limits
80/290
If f(x) does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40
Some Remarks
7/29/2019 Lecture 2 - Limits and One-Sided Limits
81/290
If f(x) does not approach any
particular real number as x
approaches a, then we say
limxa f(x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
(Heaviside Function)
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
limx0
H(x) dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 40
For today
7/29/2019 Lecture 2 - Limits and One-Sided Limits
82/290
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 16 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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Theorem
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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Theorem
If limxa f(x) exists, then it is unique.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40
Limit Theorems
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85/290
Theorem
If limxa f(x) exists, then it is unique.
If c R , then limxa c
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
86/290
Theorem
If limxa f(x) exists, then it is unique.
If c R , then limxa c = c.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
87/290
Theorem
If limxa f(x) exists, then it is unique.
If c R , then limxa c = c.limxa x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
88/290
Theorem
If limxa f(x) exists, then it is unique.
If c R , then limxa c = c.limxa x = a
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
89/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
92/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
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Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
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Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
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Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) = L1 L2
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40
Limit Theorems
Th
7/29/2019 Lecture 2 - Limits and One-Sided Limits
97/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) = L1 L2limxa[c f(x)] =
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40
Limit Theorems
Th
7/29/2019 Lecture 2 - Limits and One-Sided Limits
98/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) = L1 L2limxa[c f(x)] = c limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40
Limit Theorems
Th
7/29/2019 Lecture 2 - Limits and One-Sided Limits
99/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa
[f(x)g(x)] = limxa
f(x) limxa
g(x) = L1 L2limxa[c f(x)] = c limxa f(x) = cL1
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
100/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
101/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)=
limxa f(x)
limxag(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
102/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)=
limxa f(x)
limxag(x)
=L1L2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
103/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)=
limxa f(x)
limxag(x)
=L1L2
, provided L2 = 0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
104/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)=
limxa f(x)
limxag(x)
=L1L2
, provided L2 = 0
limxa (f(x))
n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
105/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)=
limxa f(x)
limxag(x)
=L1L2
, provided L2 = 0
limxa (f(x))
n =
limxa f(x)
n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Limit Theorems
Theorem
7/29/2019 Lecture 2 - Limits and One-Sided Limits
106/290
Theorem
Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .
limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim
xa[f(x)g(x)] = lim
xaf(x) lim
xag(x) = L1 L2
limxa[c f(x)] = c limxa f(x) = cL1
limxa
f(x)
g(x)=
limxa f(x)
limxag(x)
=L1L2
, provided L2 = 0
limxa (f(x))
n =
limxa f(x)
n = (L1)n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 40
Evaluate: limx1
(2x2 + 3x 4)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
107/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
108/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
109/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 +
7/29/2019 Lecture 2 - Limits and One-Sided Limits
110/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
111/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
112/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
113/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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= 2
limx1 x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
115/290
= 2
limx1 x2
+ 3
limx1 x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
116/290
= 2
limx1 x2
+ 3
limx1 x limx1 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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= 2
limx1 x2
+ 3
limx1 x limx1 4
= 2
lim
x1x
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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= 2
limx1 x2
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
119/290
= 2
limx1 x2
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
120/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
121/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2 + 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
122/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2 + 3(
1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
l2
l l
7/29/2019 Lecture 2 - Limits and One-Sided Limits
123/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2 + 3(
1)
4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
124/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2 + 3(
1)
4
= 5
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
125/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2 + 3(
1)
4
= 5
In general:
Remark
If f is a polynomial function, then limxa f(x)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
2
li2
3
li
li 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
126/290
= 2
limx1 x
+ 3
limx1 x limx1 4
= 2
lim
x1x
2+ 3
lim
x1x
lim
x14
= 2(
1)2 + 3(
1)
4
= 5
In general:
Remark
If f is a polynomial function, then limxa f(x) = f(a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
127/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
li 4x3 + 3x2 x + 1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
128/290
limx2
4x3 + 3x2 x + 1x2 + 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
li 4x3
+ 3x2
x + 1lim
x 2(4x3 + 3x2 x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
129/290
limx2
4x + 3x x + 1x2 + 2
= x2 lim
x2(x2 + 2)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
li 4x3
+ 3x2
x + 1lim
x2(4x3 + 3x2 x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
130/290
limx2
4x + 3x x + 1x2 + 2
= x2 lim
x2(x2 + 2)
=4(8)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
lim 4x3
+ 3x2
x + 1lim
x2(4x3 + 3x2 x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
131/290
limx2
4x + 3x x + 1x2 + 2
= x 2 lim
x2(x2 + 2)
=4(8) + 3(4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 + 3x2
x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
132/290
limx2
4x + 3x x + 1x2 + 2
= x 2 lim
x2(x2 + 2)
=4(8) + 3(4) (2) + 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 + 3x2
x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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limx2
4x + 3x x + 1x2 + 2
= x 2lim
x2(x2 + 2)
=4(8) + 3(4) (2) + 1
4 + 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 + 3x2
x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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limx2
4x + 3x x + 1x2 + 2
=lim
x2(x2 + 2)
=4(8) + 3(4) (2) + 1
4 + 2
= 176
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
lim 4x3
+ 3x2
x + 1 =lim
x2(4x3 + 3x2
x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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limx2
4x + 3x x + 1x2 + 2
=lim
x2(x2 + 2)
=4(8) + 3(4) (2) + 1
4 + 2
= 176
Remark
If f is a rational function and f(a) is defined,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
lim 4x3
+ 3x2
x + 12
=lim
x2(4x3 + 3x2
x + 1)
2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
136/290
limx2
4x + 3x x + 1x2 + 2 lim
x2(x2 + 2)
=4(8) + 3(4) (2) + 1
4 + 2
= 176
Remark
If f is a rational function and f(a) is defined, then lim
xaf(x) = f(a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxan
f(x) =n
limxa f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
137/290
f ( )
f ( ),
provided limxa f(x) > 0 when n is even.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxan
f(x) =n
limxa f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
138/290
f ( )
f ( ),
provided limxa f(x) > 0 when n is even.
limx3
3x
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) =
nlimxa f(x)
,
7/29/2019 Lecture 2 - Limits and One-Sided Limits
139/290
f ( )
f ( )
provided limxa f(x) > 0 when n is even.
limx3
3x
1 = limx3
(3x
1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) =
nlimxa f(x)
,
7/29/2019 Lecture 2 - Limits and One-Sided Limits
140/290
f ( )
f ( )
provided limxa f(x) > 0 when n is even.
limx3
3x
1 = limx3
(3x
1) =
8
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) =
nlimxa f(x)
,
7/29/2019 Lecture 2 - Limits and One-Sided Limits
141/290
f ( )
f ( )
provided limxa f(x) > 0 when n is even.
limx3
3x
1 = limx3
(3x
1) =
8 = 2
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) =
nlimxa f(x)
,
7/29/2019 Lecture 2 - Limits and One-Sided Limits
142/290
( )
( )
provided limxa f(x) > 0 when n is even.
limx3
3x
1 = limx3
(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) =
nlimxa f(x)
,
7/29/2019 Lecture 2 - Limits and One-Sided Limits
143/290
provided lim
xa f(x) > 0 when n is even.
limx3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) =
nlimxa f(x)
,
7/29/2019 Lecture 2 - Limits and One-Sided Limits
144/290
provided lim
xa f(x) > 0 when n is even.
limx3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf
(x
) = nlimxa f
(x
),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
145/290
provided lim
xa f(x) > 0 when n is even.
limx3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2 = 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) = nlimxa
f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
146/290
provided lim
xa f(x) > 0 when n is even.
lim
x3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2 = 1
limx7/2
4
3 2x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) = nlimxa
f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
147/290
provided lim
xa f(x) > 0 when n is even.
lim
x3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2 = 1
limx7/2
4
3 2x dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) = nlimxa
f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
148/290
provided lim
xa f(x) > 0 when n is even.
lim
x3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) = nlimxa
f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
149/290
provided lim
xa f(x) > 0 when n is even.
lim
x3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4 =??
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Theorem
Suppose limxa f(x) exists and n N . Then,
limxa
nf(x) = nlimxa
f(x),
7/29/2019 Lecture 2 - Limits and One-Sided Limits
150/290
provided lim
xa f(x) > 0 when n is even.
lim
x3
3x
1 = lim
x3(3x
1) =
8 = 2
2
limx1
3
x + 4
x 2 =3
1 + 41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4 =?? (for now)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
151/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
152/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
153/290
=
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
154/290
=
limx3
2x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
155/290
=
limx3
2x2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
156/290
=
limx3
2x2
limx3
(5x + 1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
157/290
=
limx3
2x2
limx3
(5x + 1)
limx
3(x3 x + 4)
3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
158/290
=
limx3
2x2
limx3
(5x + 1)
limx
3(x3 x + 4)
3
=
18
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
159/290
=
limx3
2x2
limx3
(5x + 1)
limx
3(x3 x + 4)
3
=
18 4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
160/290
=
limx3
2x2
limx3
(5x + 1)
limx
3(x3 x + 4)
3
=
18 4
28
3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
Evaluate: limx3
2x2 5x + 1
x3 4x 1
3
limx32x2
5x + 1
x3 4x 1 3
=
limx32x2
5x + 1
x3 x + 4 3
3
7/29/2019 Lecture 2 - Limits and One-Sided Limits
161/290
=
limx3
2x2
limx3
(5x + 1)
limx
3(x3 x + 4)
3
=
18 4
28
3
=1
8
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 40
7/29/2019 Lecture 2 - Limits and One-Sided Limits
162/290
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
7/29/2019 Lecture 2 - Limits and One-Sided Limits
163/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
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x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0
7/29/2019 Lecture 2 - Limits and One-Sided Limits
165/290
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1)
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x1
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1) = 0.
7/29/2019 Lecture 2 - Limits and One-Sided Limits
167/290
x1
x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1) = 0.
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x x
But when x = 1, 3x2 4x + 1
x
1=
(3x 1)(x 1)x
1= 3x 1.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1) = 0.
7/29/2019 Lecture 2 - Limits and One-Sided Limits
169/290
But when x = 1, 3x2 4x + 1
x
1=
(3x 1)(x 1)x
1= 3x 1.
Since we are just taking the limit as x 1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1) = 0.
7/29/2019 Lecture 2 - Limits and One-Sided Limits
170/290
But when x = 1, 3x2 4x + 1
x
1=
(3x 1)(x 1)x
1= 3x 1.
Since we are just taking the limit as x 1,
limx1
3x2 4x + 1x 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1) = 0.
7/29/2019 Lecture 2 - Limits and One-Sided Limits
171/290
But when x = 1, 3x2 4x + 1
x
1=
(3x 1)(x 1)x
1= 3x 1.
Since we are just taking the limit as x 1,
limx1
3x2 4x + 1x 1 = limx1(3x 1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
3x2 4x + 1
= 0 and lim
x1(x 1) = 0.
7/29/2019 Lecture 2 - Limits and One-Sided Limits
172/290
But when x = 1, 3x2 4x + 1
x
1=
(3x 1)(x 1)x
1= 3x 1.
Since we are just taking the limit as x 1,
limx1
3x2 4x + 1x 1 = limx1(3x 1) = 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 40
Definition
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Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0
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174/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0 then
lim
xa
f(x)
g(x)
is called an indeterminate form of type0
0.
7/29/2019 Lecture 2 - Limits and One-Sided Limits
175/290
yp0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0 then
lim
xa
f(x)
g(x)
is called an indeterminate form of type0
0.
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176/290
yp0
Remarks:
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0 then
limx
a
f(x)
g(x)
is called an indeterminate form of type0
0.
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0
Remarks:
1 If f(a) = 0 and g(a) = 0, then f(a)g(a)
is undefined!
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0 then
limx
a
f(x)
g(x)
is called an indeterminate form of type0
0.
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0
Remarks:
1 If f(a) = 0 and g(a) = 0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0 then
limx
a
f(x)
g(x)
is called an indeterminate form of type0
0.
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0
Remarks:
1 If f(a) = 0 and g(a) = 0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
3 Some techniques used in evaluating such limits are:
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Definition
If limxa f(x) = 0 and limxag(x) = 0 then
limx
a
f(x)
g(x)
is called an indeterminate form of type0
0.
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0
Remarks:
1 If f(a) = 0 and g(a) = 0, then f(a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.
3 Some techniques used in evaluating such limits are:
FactoringRationalization
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x + 1
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181/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x + 1
0
0
7/29/2019 Lecture 2 - Limits and One-Sided Limits
182/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x + 1
0
0
limx 1
x2 + 2x + 1
x + 1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
183/290
x1 x + 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x +1
0
0lim
x 1x2 + 2x + 1
x + 1= lim
x 1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
184/290
x1 x + 1 x1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x +1
0
0lim
x1x2 + 2x + 1
x + 1= lim
x1(x + 1)2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
185/290
x 1 x + 1 x 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x +1
0
0lim
x1x2 + 2x + 1
x + 1= lim
x1(x + 1)2
x + 1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
186/290
x 1 x + 1 x 1 x + 1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: lim
x1
x2 + 2x + 1
x +1
0
0lim
x1x2 + 2x + 1
x + 1= lim
x1(x + 1)2
x + 1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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x 1 x + 1 x 1 x + 1
= limx
1(x + 1)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
7/29/2019 Lecture 2 - Limits and One-Sided Limits
188/290
Examples
Evaluate: limx
1
x2 + 2x + 1
x +1
0
0lim
x1x2 + 2x + 1
x + 1= lim
x1(x + 1)2
x + 1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
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= limx
1(x + 1)
= (1 + 1)
= 0
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
190/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0
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191/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0lim
x2x3 + 8
x2 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
192/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0lim
x2x3 + 8
x2 4 = limx2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
193/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0lim
x2x3 + 8
x2 4 = limx2(x + 2)(x2 2x + 4)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
194/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
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195/290
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0lim
x2x3 + 8
x2 4 = limx2(x + 2)(x2 2x + 4)
(x + 2)(x 2)2 2 4
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196/290
= limx
2
x2 2x + 4x
2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0lim
x2x3 + 8
x2 4 = limx2(x + 2)(x2 2x + 4)
(x + 2)(x 2)2 2 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
197/290
= limx
2
x2 2x + 4x
2
=4 + 4 + 4
2 2
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx
2
x3 + 8
x2
4
0
0lim
x2x3 + 8
x2 4 = limx2(x + 2)(x2 2x + 4)
(x + 2)(x 2)2 2 + 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
198/290
= limx
2
x2 2x + 4x
2
=4 + 4 + 4
2 2= 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 40
Examples
Evaluate: limx4
x2 162x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
199/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
7/29/2019 Lecture 2 - Limits and One-Sided Limits
200/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
201/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
202/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
203/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
= limx4
(x2 16)(2 +x)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
204/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
= limx4
(x2 16)(2 +x)4 x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
205/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
= limx4
(x2 16)(2 +x)4 x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
206/290
= limx4
(x 4)(x + 4)(2 +x)4 x
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
= limx4
(x2 16)(2 +x)4 x
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= limx4
(x 4)(x + 4)(2 +x)4 x
= limx4
[(x + 4)(2 +x)]
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
= limx4
(x2 16)(2 +x)4 x
( )( )(
)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
208/290
= limx4
(x 4)(x + 4)(2 +x)4 x
= limx4
[(x + 4)(2 +x)]
= (8)(4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx4
x2 162x
0
0
limx4x2
16
2x = limx4x2
16
2x 2 +
x
2 +x
= limx4
(x2 16)(2 +x)4 x
( )( )(
)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
209/290
= limx4
(x 4)(x + 4)(2 +x)4 x
= limx4
[(x + 4)(2 +x)]
= (8)(4)
= 32
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
7/29/2019 Lecture 2 - Limits and One-Sided Limits
210/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
7/29/2019 Lecture 2 - Limits and One-Sided Limits
211/290
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8
7/29/2019 Lecture 2 - Limits and One-Sided Limits
212/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
7/29/2019 Lecture 2 - Limits and One-Sided Limits
213/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
7/29/2019 Lecture 2 - Limits and One-Sided Limits
214/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x
7/29/2019 Lecture 2 - Limits and One-Sided Limits
215/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
216/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
7/29/2019 Lecture 2 - Limits and One-Sided Limits
217/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
= limx8
x 8
7/29/2019 Lecture 2 - Limits and One-Sided Limits
218/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
= limx8
x 8(x 8)(x + 1)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
219/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
= limx8
x 8(x 8)(x + 1)( 3
x2 + 2 3
x + 4)
7/29/2019 Lecture 2 - Limits and One-Sided Limits
220/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
= limx8
x 8(x 8)(x + 1)( 3
x2 + 2 3
x + 4)
1
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221/290
=limx8
1
(x + 1)( 3x2 + 2 3x + 4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
= limx8
x 8(x 8)(x + 1)( 3
x2 + 2 3
x + 4)
1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
222/290
=limx8
1
(x + 1)( 3x2 + 2 3x + 4)=
1
9(4 + 4 + 4)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
Examples
Evaluate: limx8
3
x 2x2 7x 8
0
0
limx8
3
x 2x2 7x 8 = limx8
3
x 2x2 7x 8
3
x2
+ 23
x + 43x2 + 2 3
x + 4
= limx8
x 8(x 8)(x + 1)( 3
x2 + 2 3
x + 4)
1
7/29/2019 Lecture 2 - Limits and One-Sided Limits
223/290
=limx8
1
(x + 1)( 3x2 + 2 3x + 4)=
1
9(4 + 4 + 4)
=1
108
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 28 / 40
For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
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224/290
3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 29 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
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225/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
As x 1, the value of f(x) dependson whether x < 1 or x > 1.
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226/290
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3
1
2
3
4
0
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227/290
3
2
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
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228/290
3
2
1
As x approaches 1 through values less than 1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
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3
2
1
As x approaches 1 through values less than 1, f(x) approaches 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
7/29/2019 Lecture 2 - Limits and One-Sided Limits
230/290
3
2
1
As x approaches 1 through values less than 1, f(x) approaches 2.As x approaches 1 through values greater than 1,
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 4
Consider: f(x) =
3 5x2, x < 1
4x
3, x
1
As x 1, the value of f(x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3
1
1
2
3
4
0
7/29/2019 Lecture 2 - Limits and One-Sided Limits
231/290
3
2
1
As x approaches 1 through values less than 1, f(x) approaches 2.As x approaches 1 through values greater than 1, f(x) approaches 1.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 40
Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
2
0
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232/290
1
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 40
Illustration 5
Consider: g(x) =
x
2 1 1 2