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2.6 – Limits Involving Infinity
2
Definition
The notation
lim ( )x a
f x
means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side) but not equal to a.
a
f
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Vertical Asymptote
The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following six statements is true:
lim ( ) ______x a
f x
lim ( ) ______x a
f x
lim ( ) _______x a
f x
4
Examples
Find the limit.
1.
2. State all vertical asymptotes for the following function and write the equivalent limit statement for each asymptote.
35lim
( 5)
x
x
e
x
2
2 5( )
4 3
xf x
x x
5
Definition
Let f be a function defined on some interval (a, ∞). Then
lim ( )x
f x L
means that the value of f (x) can be made as close to L as we like by taking x sufficiently large.
L
f
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Horizontal Asymptote
The line y = L is called a horizontal asymptote of the curve y = f(x) if either
lim ( )x
f x L
lim ( )x
f x L
or
7
Examples
4. lim x
xe
01. lim ln
xx
Evaluate the following. State the equations of any asymptotes that result from the limit.
2
2. lim tanx
x
15. lim tanx
x
33. lim
( 5)
x
x
e
x
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Algebra Review
1. Simplify
2. Bring the expression into the radical and simplify.
34
3 3
1/2
3 1 1/
xx
x x x
331/ 3 2y y y
n n na b ab
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Properties
lim 0nx
a
x If n is a positive integer, then ,
where a is some constant.
To evaluate limits going to infinity, we often use the technique of multiplying the expression by 1 in the form of
1/
1/
n
n
x
x
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Examples
Evaluate the limit and determine any asymptotes.
1. 2.
3.
2
2
2lim
3 1t
t
t t
2
2lim
9 1y
y
y
2
2
sinlimx
x
x
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You Try It
Evaluate the limit and use the results to state any asymptotes that exist.
1. 2.
3. 4.
3
5
7 3 2lim
4 3p
p p
p
2 2lim y
ye
y
2
1lim
25x
x
x x
2
3lim
2x
x
x
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