1
Section 1.8: Infinite Limits
Infinite Limits
1
.
x 1 x  1
Suppose we want to find lim
x
0
y  f ( x) 
1
x 1
0.9
0.99
0.999
0.9999
2
1.0001
1.001
.01
1.1
2
As x → 1 left, y 
-100
-1000
-10000
undefined
10000
1000
100
10
1
1

x 1
As x → 1 right, y 
1

x 1
2
Note: A limit in which y  f (x) increases or decreases without bound as x approaches a
fixed value is called an infinite limit.
Example 1: Determine lim
x0
1
x2
.
Solution:
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3
Example 2: Determine lim
x 1
2 x
.
1 x
Solution:
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4
Example 3: Determine lim csc x .
x 
1
1
 is undefined, so direct substitution fails. To
sin  0
1
determine the behavior of f ( x)  csc x 
as x approaches   3.141592654 from the
sin x
right, a plot chart can be helpful.
Solution: Note that csc  
x
3.2
3.15
3.142
3.1416
3.141593
1
sin x
csc(3.2)  1 / sin( 3.2)  -17.13087236
y  f ( x)  csc x 
-118.9449955
-2454.913073
-136120.9042
-2886750.969
Thus, it can be seen that x gets closer to  from the right, the functional values are
increasing infinitely in the negative direction without bound. Thus, we conclude that
lim csc x  
x 
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Vertical Asymptotes
A vertical asymptote is a vertical line of the form x = a that the graph of a function
approaches infinity or negative infinity as x gets arbitrarily close to a. The graph of
y  f (x) increases infinitely without bound in the positive or negative direction
(approaches positive or negative infinity) as x approaches a, then x = a is a vertical
asymptote.
For example,
f ( x) 
1
has a vertical asymptote at x = 1.
x 1
f ( x) 
1
x2
has a vertical asymptote at x = 0.
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Finding Vertical Asymptotes Algebraically
f ( x)
if f (a)  0 and g (a)  0 ,
g ( x)
that is, a value x = a where the Numerator  0 and Denominator = 0.
A value x = a is a vertical asymptote for the function h( x) 
Note: If there is a value x = a where both f (a)  0 and g (a)  0 , that is, a value x = a where
both the Numerator  0 and Denominator = 0, then look for a removable discontinuity.
Example 4: Find the vertical asymptotes for the function f ( x) 
x2
x2 1
.
Solution:
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Example 5: Find the vertical asymptotes for the function f ( x)  csc x .
1
. The denominator
sin x
sin x  0 when x   ,  3 ,  2 ,   , 0,  , 2 , 3 ,  , that is, integer multiples of  .
Since the numerator of this function 1  0 for any value of x, the vertical asymptotes are all
the integer multiples of  , that is
Solution: For this problem, start by writing f ( x)  csc x 
Vertical Asymptotes: x   n , where n is an integer.
These vertical asymptotes can be seen by the following graph.
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Example 6: Find the vertical asymptotes for the function f ( x) 
x2
x2  4
.
Solution:
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