Lecture 2 - Limits and One-Sided Limits

7/29/2019 Lecture 2 - Limits and One-Sided Limits

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Limit of a Function and

One-sided limits

Mathematics 53

Institute of Mathematics (UP Diliman)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 1 / 40


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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


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


R ,

what is the value of f at x near a,



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Introduction


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


x f(x)



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Illustration 1


x f(x)

0



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Illustration 1


x f(x)

0 1



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Illustration 1


x f(x)

0 10.5



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Illustration 1


x f(x)

0 10.5 0.5



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Illustration 1


x f(x)

0 10.5 0.5

0.9



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Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7



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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



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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

x f(x)



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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

x f(x)

2



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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

x f(x)

2 5



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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

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


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


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.


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


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


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


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


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


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


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


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


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


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


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.


Illustration 2


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Illustration 2

Consider: g(x) =3x2 4x + 1

x 1


Illustration 2


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Illustration 2


x 1=

(3x 1)(x 1)x 1


Illustration 2


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Illustration 2


x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1


Illustration 2


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x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4


Illustration 2


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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,


Illustration 2


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x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4



Illustration 2


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x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4



Illustration 2


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x 1=

(3x 1)(x 1)x 1

= 3x 1, x = 1

1 1 2 31

1

2

3

4



Illustration 2


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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.


Illustration 3


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Consider: h(x) =

3x 1, x = 10, x = 1


Illustration 3


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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4


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



Illustration 3


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Consider: h(x) =

3x 1, x = 10, x = 1

1 1 2 31

1

2

3

4



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.


Limit


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Limit


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Intuitive Notion of a Limit


Limit


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Intuitive Notion of a Limita R , L R


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


Limit


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The limit of f(x) as x approaches a is L


Limit


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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.


Limit


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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


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4limx1

(3x 1)


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4limx1

(3x 1) = 2


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)


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).


Examples


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g(x) =3

x

2

4

x +1

x 1

1 1 2 31

1

2

3

4


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


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


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.


Examples


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h(x) =

3x

1, x

= 1

0, x = 1

1 1 2 31

1

2

3

4


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)


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


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).


Some Remarks


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Remark

In finding limxa f(x):


Some Remarks


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Remark


We only need to consider values of x very close to a but not exactly at a.


Some Remarks


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Remark



Thus, limxa f(x) is NOT NECESSARI LY the same as f(a).


Some Remarks


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Remark



Thus, limxa f(x) is NOT NECESSARI LY the same as f(a).

We let x approach a from BOTH SIDES of a.


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).


Some Remarks


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Iff(x)

does not approach any




e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)


Some Remarks


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If f(x) does not approach any




e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0



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Some Remarks


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e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.


Some Remarks


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e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1?


Some Remarks


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e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1? No.


Some Remarks


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e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1? No.

limx0

H(x) dne


For today


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2 Evaluating Limits

3 One-sided Limits


Limit Theorems


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Theorem


Limit Theorems


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Theorem

If limxa f(x) exists, then it is unique.


Limit Theorems


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Theorem


If c R , then limxa c


Limit Theorems


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Theorem


If c R , then limxa c = c.


Limit Theorems


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Theorem


If c R , then limxa c = c.limxa x


Limit Theorems


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Theorem


If c R , then limxa c = c.limxa x = a


Limit Theorems


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Theorem

Suppose limxa f(x) = L1 and limxag(x) = L2. Let c R , n N .


Limit Theorems


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Theorem


limxa[f(x)g(x)]


Limit Theorems


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Theorem


limxa[f(x)g(x)] = limxa f(x)


Limit Theorems


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Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x)


Limit Theorems


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Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2


Limit Theorems


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Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)]


Limit Theorems


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Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x)


Limit Theorems


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Theorem



[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


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Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2limxa[c f(x)] =


Limit Theorems

Th


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Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2limxa[c f(x)] = c limxa f(x)


Limit Theorems

Th


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Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) = L1 L2limxa[c f(x)] = c limxa f(x) = cL1


Limit Theorems

Theorem


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Theorem


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)


Limit Theorems

Theorem


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Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2


limxa

f(x)

g(x)=

limxa f(x)

limxag(x)


Limit Theorems

Theorem


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Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2


limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2


Limit Theorems

Theorem


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Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2


limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0


Limit Theorems

Theorem


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Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2


limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

limxa (f(x))

n


Limit Theorems

Theorem


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Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2


limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

limxa (f(x))

n =

limxa f(x)

n


Limit Theorems

Theorem


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Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) = L1 L2


limxa

f(x)

g(x)=

limxa f(x)

limxag(x)

=L1L2

, provided L2 = 0

limxa (f(x))

n =

limxa f(x)

n = (L1)n


Evaluate: limx1

(2x2 + 3x 4)


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4)


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 +


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4


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Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4


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= 2

limx1 x2


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4


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= 2

limx1 x2

+ 3

limx1 x


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4


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= 2

limx1 x2

+ 3

limx1 x limx1 4


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2


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= 2

limx1 x2

+ 3

limx1 x limx1 4

= 2

lim

x1x

2


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2


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= 2

limx1 x2

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2


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= 2

limx1 x2

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2


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= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2


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= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2


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= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

l2

l l


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= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

4


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4


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= 2

limx1 x

+ 3

limx1 x limx1 4

= 2

lim

x1x

2+ 3

lim

x1x

lim

x14

= 2(

1)2 + 3(

1)

4

= 5


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4


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= 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)


Evaluate: limx1

(2x2 + 3x 4)

limx1

(2x2 + 3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4


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= 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).


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2


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Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

li 4x3 + 3x2 x + 1


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limx2

4x3 + 3x2 x + 1x2 + 2


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

li 4x3

+ 3x2

x + 1lim

x 2(4x3 + 3x2 x + 1)


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limx2

4x + 3x x + 1x2 + 2

= x2 lim

x2(x2 + 2)


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

li 4x3

+ 3x2

x + 1lim

x2(4x3 + 3x2 x + 1)


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limx2

4x + 3x x + 1x2 + 2

= x2 lim

x2(x2 + 2)

=4(8)


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1lim

x2(4x3 + 3x2 x + 1)


131/290

limx2

4x + 3x x + 1x2 + 2

= x 2 lim

x2(x2 + 2)

=4(8) + 3(4)


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)


132/290

limx2

4x + 3x x + 1x2 + 2

= x 2 lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)


133/290

limx2

4x + 3x x + 1x2 + 2

= x 2lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

4 + 2


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)


134/290

limx2

4x + 3x x + 1x2 + 2

=lim

x2(x2 + 2)

=4(8) + 3(4) (2) + 1

4 + 2

= 176


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 + 3x2

x + 1)


135/290

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,


Evaluate: limx2

4x3 + 3x2 x + 1x2 + 2

lim 4x3

+ 3x2

x + 12

=lim

x2(4x3 + 3x2

x + 1)

2


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).


Theorem

Suppose limxa f(x) exists and n N . Then,

limxan

f(x) =n

limxa f(x),


137/290

f ( )

f ( ),

provided limxa f(x) > 0 when n is even.


Theorem


limxan

f(x) =n

limxa f(x),


138/290

f ( )

f ( ),


limx3

3x

1


Theorem


limxa

nf(x) =

nlimxa f(x)

,


139/290

f ( )

f ( )


limx3

3x

1 = limx3

(3x

1)


Theorem


limxa

nf(x) =

nlimxa f(x)

,


140/290

f ( )

f ( )


limx3

3x

1 = limx3

(3x

1) =

8


Theorem


limxa

nf(x) =

nlimxa f(x)

,


141/290

f ( )

f ( )


limx3

3x

1 = limx3

(3x

1) =

8 = 2

2


Theorem


limxa

nf(x) =

nlimxa f(x)

,


142/290

( )

( )


limx3

3x

1 = limx3

(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2


Theorem


limxa

nf(x) =

nlimxa f(x)

,


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


Theorem


limxa

nf(x) =

nlimxa f(x)

,


144/290

provided lim


limx3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2


Theorem


limxa

nf

(x

) = nlimxa f

(x

),


145/290

provided lim


limx3

3x

1 = lim

x3(3x

1) =

8 = 2

2

limx1

3

x + 4

x 2 =3

1 + 41 2 = 1


Theorem


limxa

nf(x) = nlimxa

f(x),


146/290

provided lim


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


Theorem


limxa

nf(x) = nlimxa

f(x),


147/290

provided lim


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


Theorem


limxa

nf(x) = nlimxa

f(x),


148/290

provided lim


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


Theorem


limxa

nf(x) = nlimxa

f(x),


149/290

provided lim


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 =??


Theorem


limxa

nf(x) = nlimxa

f(x),


150/290

provided lim


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)


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3


151/290


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3


152/290


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3


153/290

=


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3


154/290

=

limx3

2x2


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3


155/290

=

limx3

2x2


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3


156/290

=

limx3

2x2

limx3

(5x + 1)


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3


157/290

=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3


158/290

=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3


159/290

=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18 4


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3


160/290

=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18 4

28

3


Evaluate: limx3

2x2 5x + 1

x3 4x 1

3

limx32x2

5x + 1

x3 4x 1 3

=

limx32x2

5x + 1

x3 x + 4 3

3


161/290

=

limx3

2x2

limx3

(5x + 1)

limx

3(x3 x + 4)

3

=

18 4

28

3

=1

8



162/290


x 1 . From earlier, limx1g(x) = 2.

Can we arrive at this conclusion computationally?


163/290





Note that limx1

3x2 4x + 1


164/290

x1





Note that limx1

3x2 4x + 1

= 0


165/290

x1





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1)


166/290

x1

x1





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.


167/290

x1

x1





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.


168/290

x x

But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.


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,





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.


170/290

But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.


limx1

3x2 4x + 1x 1





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.


171/290

But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.


limx1

3x2 4x + 1x 1 = limx1(3x 1)





Note that limx1

3x2 4x + 1

= 0 and lim

x1(x 1) = 0.


172/290

But when x = 1, 3x2 4x + 1

x

1=

(3x 1)(x 1)x

1= 3x 1.


limx1

3x2 4x + 1x 1 = limx1(3x 1) = 2.


Definition


173/290


Definition

If limxa f(x) = 0 and limxag(x) = 0


174/290


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.


175/290

yp0


Definition


lim

xa

f(x)

g(x)


0.


176/290

yp0

Remarks:


Definition


limx

a

f(x)

g(x)


0.


177/290

0

Remarks:

1 If f(a) = 0 and g(a) = 0, then f(a)g(a)

is undefined!


Definition


limx

a

f(x)

g(x)


0.


178/290

0

Remarks:


is undefined!

2 The limit above MAY or MAY NOT exist.


Definition


limx

a

f(x)

g(x)


0.


179/290

0

Remarks:


is undefined!


3 Some techniques used in evaluating such limits are:


Definition


limx

a

f(x)

g(x)


0.


180/290

0

Remarks:


is undefined!


3 Some techniques used in evaluating such limits are:

FactoringRationalization


Examples

Evaluate: lim

x1

x2 + 2x + 1

x + 1


181/290


Examples

Evaluate: lim

x1

x2 + 2x + 1

x + 1

0

0


182/290


Examples

Evaluate: lim

x1

x2 + 2x + 1

x + 1

0

0

limx 1

x2 + 2x + 1

x + 1


183/290

x1 x + 1


Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x 1x2 + 2x + 1

x + 1= lim

x 1


184/290

x1 x + 1 x1


Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2


185/290

x 1 x + 1 x 1


Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2

x + 1


186/290

x 1 x + 1 x 1 x + 1


Examples

Evaluate: lim

x1

x2 + 2x + 1

x +1

0

0lim

x1x2 + 2x + 1

x + 1= lim

x1(x + 1)2

x + 1


187/290

x 1 x + 1 x 1 x + 1

= limx

1(x + 1)



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


189/290

= limx

1(x + 1)

= (1 + 1)

= 0


Examples

Evaluate: limx

2

x3 + 8

x2

4


190/290


Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0


191/290


Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4


192/290


Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2


193/290


Examples

Evaluate: limx

2

x3 + 8

x2

4

0

0lim

x2x3 + 8

x2 4 = limx2(x + 2)(x2 2x + 4)


194/290



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


196/290

= limx

2

x2 2x + 4x

2


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


197/290

= limx

2

x2 2x + 4x

2

=4 + 4 + 4

2 2


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


198/290

= limx

2

x2 2x + 4x

2

=4 + 4 + 4

2 2= 3


Examples

Evaluate: limx4

x2 162x


199/290


Examples

Evaluate: limx4

x2 162x

0

0


200/290


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x


201/290


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x


202/290


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x


203/290


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)


204/290


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x


205/290


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x


206/290

= limx4

(x 4)(x + 4)(2 +x)4 x


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x


207/290

= limx4

(x 4)(x + 4)(2 +x)4 x

= limx4

[(x + 4)(2 +x)]


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

( )( )(

)


208/290

= limx4

(x 4)(x + 4)(2 +x)4 x

= limx4

[(x + 4)(2 +x)]

= (8)(4)


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2 +

x

2 +x

= limx4

(x2 16)(2 +x)4 x

( )( )(

)


209/290

= limx4

(x 4)(x + 4)(2 +x)4 x

= limx4

[(x + 4)(2 +x)]

= (8)(4)

= 32


Examples

Evaluate: limx8

3

x 2x2 7x 8


210/290



211/290

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8


212/290


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8


213/290


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2


214/290


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


215/290


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


216/290


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


217/290


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


218/290


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)


219/290


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)


220/290


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


221/290

=limx8

1

(x + 1)( 3x2 + 2 3x + 4)


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


222/290

=limx8

1

(x + 1)( 3x2 + 2 3x + 4)=

1

9(4 + 4 + 4)


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


223/290

=limx8

1

(x + 1)( 3x2 + 2 3x + 4)=

1

9(4 + 4 + 4)

=1

108


For today


2 Evaluating Limits


224/290

3 One-sided Limits


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


225/290


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.


226/290


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


227/290

3

2

1


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


228/290

3

2

1

As x approaches 1 through values less than 1,


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


229/290

3

2

1

As x approaches 1 through values less than 1, f(x) approaches 2.


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


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,


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


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.


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0


232/290

1


Illustration 5

Consider: g(x) =

x

2 1 1 2

Lecture 2 - Limits and One-Sided Limits

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