Section 1.5: Infinite Limits
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Section 1.5: Infinite Limits
Vertical AsymptoteIf f(x) approaches infinity (or negative infinity) as x
approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f.
Infinite LimitsThe limit statement such as
means that the function f increases without bound as x approaches c from either side, while
means that the function g decreases without bound as x approaches c from either side.
limx c
f x
limx c
f x
21x
h x
21x
g x
Example 1
Sketch a graph of a function with the following characteristics:
The graph has discontinuities at x = -2, 0, and 3. Only x = 0 is removable.
limx 2
f x
limx 2
f x
Example 2 Use the graph and complete the table to find the limit (if it exists).
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x) -100 -10000 -1000000 -100-10000-1000000
If the function behaves the same around an asymptote, then the infinite limit exists.
The function decreases without bound as x approaches 2 from
either side.
limx 2
1
x 2 2
DNE
limx 2
1
x 2 2
Example 3 Use the graph and complete the table to find the limit (if it exists).
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x)
If the function behaves the different around an asymptote, then the infinite limit does not
exist.The function increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from
the left.
limx 2
1x 2
limx 2
1x 2
DNE
-10 -100 -1000 101001000DNE
Example 4 Use the graph and complete the table to find the limits (if they exist).
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x)
One-Sided Infinite Limits do Exist
The function increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from
the left.
limx 2
1x 2
limx 2
1x 2
limx 2
1x 2
limx 2
1x 2
-10 -100 -1000 101001000DNE
The Existence of a Vertical Asymptote
If is continuous c around and g(x) ≠ 0 around c,
then x = c is a vertical asymptote of h(x) if f(c) ≠ 0 and
g(c) = 0.
h x f x g x
Big Idea: x = c is a vertical asymptote if c ONLY makes
the denominator zero.
Ex: Determine all vertical asymptotes of .
f x 2x 3x 2 1
When is the denominator zero:
x 2 1 0
x 1 x 1 0
x 1
Do the x’s make the numerator 0?
2 1 3 1
2 1 3 5
x=1 and x=-1 are vertical asymptotes
Must have equations for asymptotes
No for both
Example 2Determine all vertical asymptotes of .
f x x1x 2 x 2
When is the denominator zero:
x 2 x 2 0
x 1 x 2 0
x 1 or x 2
Do the x’s make the numerator 0?
11 0
2 1 3 No!
Yes…
x=2 is a vertical asymptote
EXTRA: What about x = -1?
f x x1x 2 x 2
11 2x
x x
12x
Therefore, x=1 is a removable discontinuity
Example 3Analytically determine all vertical asymptotes of
f x csc 2x
When is the denominator zero:
sin 2x 0
2x 0 or 2x
Do the x’s make the numerator 0?
No, since the numerator is a
constant.
We Know:
f x csc 2x 1sin 2x
x 0 or x 12
0 and π are angles that make sine 0
Period Find all of the values since trig functions
are cyclic
x n or x 12 n
where n is any integer
x n or x 12 n
(where n is any integer)
are all of the vertical asymptotes
22 1
Example 3 Cont.
Check with the graph
Analytically determine all vertical asymptotes of
f x csc 2x
Properties of Infinite LimitsLet c and L be real numbers and f and g be functions such
that:
1. Sum/Difference:
2. Product:
3. Quotient:
Example: Since , then
g c 0
limx 0
1=1 and limx 0
1x 2 =
limx 0
1 1x 2 =
limx c
f x
limx c
g x L
limx c
f x g x
limx c
f x g x , L 0
limx c
f x g x , L 0
limx c
g x f x 0
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