Limits & Continuity 1 of 10 Limits Definition: Limits describe the output, y or f(x), of a function as the input, x, approaches some particular value. L x f c x = → ) ( lim “The limit of f(x) as x approaches c equals L.” 2 ) ( lim ) ( lim ) ( lim 1 1 1 = = = → → → x h x g x f x x x
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Limits & Continuity 1 of 10
Limits
Definition: Limits describe the output, y or f(x), of a function as the input, x, approaches some particular value.
Lxfcx
=→
)(lim “The limit of f(x) as x approaches c equals L.”
2)(lim)(lim)(lim111
===→→→
xhxgxfxxx
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Limits & Continuity 2 of 10
Limits can be approached from either the left or the right side.
Right-Hand Limit
)(lim xfcx +→
“The limit of f(x) as x approaches c from the right.”
Left-Hand Limit
)(lim xfcx −→
“The limit of f(x) as x approaches c from the left.”
If the limit approached from the left and the right are equal, than the limit as x approaches c exist.
DNExf
xfxf
cx
xx
=
==
+
−+
→
→→
)(lim
2)(lim 1)(lim11
Lxf
xfxf
cx
cxcx
=
=
+
−+
→
→→
)(lim
)(lim)(lim
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Limits & Continuity 3 of 10
Algebra Approach
Substitution: Substitute directly the value of x. 15)1723(lim 23
1−=−−+
→xxx
x
Simplification: If 0/0 is the answer from substitution, use algebra to manipulate the function and find the limit. Some problems are similar to those below.
Factor
23
)1)(1()2)(1(lim
00
12lim
1
2
2
1
=+−
+−
=−
−+
→
→
ttttttt
t
t
34
)1)(4(lim
4944lim
00
49)2(lim
4
2
4
2
4
=−
−−−
−++
=−
−+
→
→
→
xxxx
xxxx
xx
x
x
x
Multiply Denominator
41
)2(21lim
)2(222lim
)2)(2(2)2)(2(1
2)2)(2(1
lim
002
121
lim
00
0
0
=+
=+
−+
++
+−
+
=+−
→→
→
→
xxxx
xxxxx
xx
xx
x
x
Multiply Root
41
241lim
2444lim
242424lim
0024lim
00
0
0
=++
=++
−+++
++⋅
−+
=−+
→→
→
→
hhhh
hh
hhhh
hh
h
h
Separate
8)sin(53lim
00)sin(53lim
0
0
=+
=+
→
→
xx
xx
xxx
x
x
yanother wa find00
asy.) (vertical undefined0#
0#0
=
=
=
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Limits & Continuity 4 of 10
Rationalization Often, but not always, used in limits that approach infinity or negative infinity.
1sinlim0
=→ x
xx
DNExx
x=
→
coslim0
1sin
lim0
=→ x
xx
0cos
lim0
=→ x
xx
1sinlim ±=±∞→
xx
1sinlim ±=±∞→
xx
x
x
n
xxaxx
±∞→±∞→±∞→<< limlimsinlim
Numerical Approach Create a table with x values that approaches closer and closer to c.
5.016385lim 24
23
0−=
−
+→ xx
xxx
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Limits & Continuity 5 of 10
Vertical Asymptote
If the limit of f(x) as x approaches a from the left or the right equals positive (or negative infinity), than there is a vertical asymptote at x = a.
To find Vertical Asymptote: 1. Factor and simplify the expression. 2. Set the denominator equal to zero.
1.
21
)2)(2()2(1
42)( 2
+
−=
−+
−−=
−
−=
x
xxx
xxxf
2.
202−=
=+
xx
±∞=±→
)(lim xfax
∞=
−∞=
−
+
−→
−→
)(lim
)(lim
2
2
xf
xf
x
x
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Limits & Continuity 6 of 10
Horizontal Asymptote If the limit of f(x) as x infinity (or negative infinity) equals a constant b, than there is a horizontal asymptote at y = b.
To find Horizontal Asymptote: 1. Take the term with the largest exponent from the top and bottom. 2. Reduce. Note: trig < powers < exponential
a. If x is left on the top, then no horizontal asymptote. b. If x cancels, then the y = the coefficients. c. If x is left on the bottom, then y = 0.
1.
2
2
22
2
2 48
438
42)(
xx
xx
xx
xxxf
−+
−=
−+
−
−=
2. 220
481
−=−=−−=x
y
bxfx
=±∞→
)(lim
x
x
n
xxaxx
±∞→±∞→±∞→<< limlimsinlim
2)(lim
2)(lim
−=
−=
−∞→
∞→
xf
xf
x
x
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Limits & Continuity 7 of 10
End Behavior Model For numerically large values of x, we can sometimes model the behavior of a complicated function by a simpler on that acts virtually the same way. To find the end behavior model, take the largest term from the numerator and denominator and reduce.
Small Values of x
Original Function End Behavior Model
Large Values of x
2124)(
3
−+−
=xxxxf 24)( xxf =
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Limits & Continuity 8 of 10
Continuity Interior Point: A function is continuous at an
interior point c of its domain if )()(lim cfxf
cx=
→
Endpoint: A function is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if
)()(lim afxfax
=+→
or )()(lim bfxfbx
=−→
Note: For a piecewise function to be continuous, both pieces of the function must equal and defined at the split a.
⎩⎨⎧
<
≥=
axxgaxxf
xh),(),(
)(
)()(lim)(lim afagafaxax
==−+ →→
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Limits & Continuity 9 of 10
Types of Discontinuity Removable One point is removed.
Infinite Limit equals to infinity.
Jump Right and left limts do not equal.
Oscillating Function oscillates with no limit.
2
1)(x
xf = ⎟⎠
⎞⎜⎝
⎛=x
xf 1sin)(
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Limits & Continuity 10 of 10
Continuity To determine if a function is continuous look at its domain. Some common domains are listed below.
Rational Functions (Fractions)
421)(−
=x
xf
All real numbers except values of x that a make the denominator equal to zero.
042 ≠−x D: 2≠x or ),2()2,( ∞∪−∞
Even Root Functions
xxf 312)( −= All Positive Numbers & Zero
0123 ≥+x D: 4≤x or ]4,(−∞
Logarithmic Functions )4(log)( 3 += xxf
Only Positive Numbers 04 >+x
D: 4−>x or ),4[ ∞− Inverse Trigonometric Functions
xxfxxf
1
1
cos)()4(sin)(
−
−
=
+=
The inside values of the inverse trig must be between –1 and 1.
141 ≤+≤− x D: 35 −≤≤− x or ]3,5[ −−
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