Section 2.6 Limits at infinity; horizontal asymptotes
Definition Let f be a function defined on (a, ∞). Then
lim f (x) = L
x→∞
means that we can make values of f (x) arbitrary close to L by taking x to be sufficiently large.
Definition Let f be a function defined on (−∞, a). Then
lim f (x) = L
x→−∞
means that we can make values of f (x) arbitrary close to L by taking x to be sufficiently large
negative.
Definition The line y = L is called a horizontal asymptote of the curve y = f (x) if
either
lim f (x) = L
x→∞
or
lim f (x) = L
x→−∞
Limit laws Suppose that c is a constant and the limits lim f (x) and lim g(x) exist. Then
x→±∞
1. lim [f (x) + g(x)] = lim f (x) + lim g(x)
x→±∞
x→±∞
x→±∞
2. lim [f (x) − g(x)] = lim f (x) − lim g(x)
x→±∞
x→±∞
x→±∞
3. lim cf (x) = c lim f (x)
x→±∞
x→±∞
4. lim f (x)g(x) = lim f (x) · lim g(x)
x→±∞
x→±∞
x→±∞
lim f (x)
f (x)
x→±∞
5. lim
=
if lim g(x) ̸= 0
lim g(x) x→±∞
x→±∞ g(x)
x→±∞
[
n
6. lim [f (x)] =
x→±∞
]n
lim f (x) where n is a positive integer
x→±∞
7. lim c = c
x→±∞
8. lim
x→±∞
√
√
n
f (x) = n lim f (x) where n is a positive integer
x→±∞
Theorem If r > 0 is a rational number, then
1
1
= lim r = 0
r
x→−∞ x
x→∞ x
lim
Example 1. Find each of the following limits:
1
x→±∞
7y 3 + 4y
y→∞ 2y 3 − y 2 + 3
(a.) lim
x+4
x→∞ x3 − 3
(b.) lim
t2 − 3t + 1
t→∞
2t + 3
(c.) lim
 an
 bm , if n = m
an x + an−1 x
+ . . . + a1 x + a0
lim
=
if n < m
x→∞ bm xm + bm−1 xm−1 + . . . + b1 x + b0
 0,
∞, if n > m
n
n−1
Example 2.√ Evaluate the following limits:
x2 + 4x
(a.) lim
x→∞ 4x + 1
(b.) lim sin x
x→∞
(c.) lim sin
x→∞
1
x
sin2 x
x→∞ x2
(d.) lim
2
1 − cos x
x→∞
x2
(e.) lim
√
(f.) lim ( x2 + 3x + 1 − x)
x→∞
Example 3. Find the horizontal and vertical asymptotes of each curve
x2 + 4
(a.) y = 2
x −1
x3 + 1
(b.) y = 2
x +x
3