Chapter 3.3 Notes
1
(c) Epstein, 2014
3.3 Limits at Infinity
Let f be a function defined on some interval (a, ¥) . Then
lim f ( x) = L
x¥
means that the values of f ( x ) can be made arbitrarily close to L by taking
x to be sufficiently large.
Let f be a function defined on some interval (-¥, a ) . Then
lim f ( x) = L
x-¥
means that the values of f ( x ) can be made arbitrarily close to L by taking
x to be sufficiently large negative.
EXAMPLE 1
Determine the limits at infinity for the following functions
x
1
-20
4
2
20
x
-20
-15
-10
-5
5
10
15
20
-2
y
-8
a)
-4
b)
y
20
-4
-3
-2
-1
sin(x)
x
0
-5
0
1
2
3
4
y
1
5
-20
-p
-40
p
2p
3p
-60
c).
-80
d)
-1
1
=0
x¥ x r
If r > 0 is a rational number such that x r is defined for all x, then
1
lim r = 0
x-¥ x
If r > 0 is a rational number, then lim
Chapter 3.3 Notes
2
(c) Epstein, 2014
Example: Evaluate the following limits and justify each step
7 x3 + 4 x
a) lim 3
x¥ 2 x - x 2 + 3
x 3 -1
b) lim 4
x¥ x + 1
ìï 0
if deg(p ) < deg(q )
ï
p ( x) ï
lim f ( x) = lim
= í L ¹ 0 if deg(p ) = deg(q )
x¥
x¥ q ( x )
ïï
ïïî DNE if deg(p ) > deg(q )
Chapter 3.3 Notes
3
Example: Find the following limits
2 x 2 -1
x + 8x2
a) lim
x¥
6t 2 + 5t
b) lim
x-¥ (1 - t )(2t - 3)
x4 + 2x + 3
c) lim
x-¥ x x 2 -1
(
)
x2 + 4x
4 x +1
d) lim
x-¥
e) lim
x¥
(
x 2 + 3x + 1 - x
)
(c) Epstein, 2014
Chapter 3.3 Notes
4
(c) Epstein, 2014
The line y = L is called a horizontal asymptote of the curve y = f ( x) if
either
lim f ( x) = L or lim f ( x) = L
x¥
x-¥
Example: Find the horizontal and vertical asymptotes of each curve.
x -9
x2 + 4
b) y =
a) y = 2
x -1
4 x 2 + 3x + 2
lim e- x = 0
x¥
Example: Given N (t ) =
50
, find lim N (t ) and graph N (t )
t ¥
1 + 3e-2t