2.6 Limits at Infinity. |x|x As x approaches infinity f(x) ? as x gets larger and larger f(x) ? as...
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Transcript of 2.6 Limits at Infinity. |x|x As x approaches infinity f(x) ? as x gets larger and larger f(x) ? as...
2.6 Limits at Infinity
|x
As x approaches infinity
f(x) ? as x gets larger and larger
f(x) ? as x gets larger and larger in the negative direction
8
f (x)
x → ∞
8)(lim
xfx
lim ( )x
f x
Fun Fact: A student, who is an excellent pattern-observer, was asked to find the following limit. He quickly answered
)(lim7
xfx
7
|x
Limit at infinity
If f(x) approaches L as x approaches ∞ or - ∞, then
the limit as x approaches a of f(x) is L.
L
f (x)
x → ∞
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.
Graphical Approach
)(lim xfx
)(lim2
xfx
)(lim2
xfx
)2(f
0
dne undefined
Vertical Asymptote:x = -2
Horizontal Asymptote:y = 0
Horizontal Asymptote
The line y = L is called a horizontal asymptote of the curve f (x) if either
lim ( )x
f x L
lim ( )x
f x L
or
L
f
Examples
Find the limits.
xx
3lim
5
3lim
2xx
5
3lim
5xx
lim 0x
xe
Horizontal Asymptote
lim 0x
xe
Since
f(x) = ex has a horizontal asymptote y = 0
Polynomial functions have no horizontal asymptotes.
Find all horizontal asymptotes of the function.
xexf
3
5)( xxg arctan)(
Finding limits at infinity
lim 0nx
c
x If n is a positive number, then
To find limits at infinity, we multiply the expression by
where n is the highest power in the denominator.
Then use the above property to simplify.
1/
1/
n
n
x
x
Examples
Find the limits.
2
2
2lim
3 1t
t
t t
xx
xx 3
52lim
3
13
45lim
4
25
xx
xxx
2
2
2lim
3 1t
t
t t
2
2lim
9 1y
y
y
2
2
sinlimx
x
x