CHAPTER 2. LIMITS
2.6
18
Limits at infinity
Definition. We write
�
�
the y-values of f (x) become infinitely close to L
as the x-values become infinitely large.
lim f (x) = L to mean that
x→∞
This limit goes by another name: f (x) has a horizontal asymptote, on the right, of
y = L.
We have obvious variations
lim f (x) = L, lim f (x) = ∞, lim f (x) = −∞,
x→−∞
x→∞
x→∞
lim f (x) = −∞
x→−∞
Fact. Here are all the basic functions you know with horizontal asymptotes (as well
as three functions that don’t have them).
1
• Suppose that p is any positive number. Then lim p = 0. In other words:
x→±∞ x
a horizontal asymptote of y = 0.
• lim ex = 0. In other words: a left-hand horizontal asymptote of y = 0.
x→−∞
• lim tan−1 (x) = π/2,
x→∞
lim tan−1 (x) = −π/2. In other words: a left-hand
x→−∞
horizontal asymptote of y = −π/2 and a right-hand horizontal asymptote of
y = π/2.
√
• lim ln(x) = ∞,
lim ex = ∞,
lim x = ∞. In other words: no
x→∞
x→∞
x→∞
horizontal asymptotes.
Now we generalize two of the above facts
Fact. If lim f (x) = L (with L �= ±∞) and lim g(x) = ±∞, then
x→a
x→a
f (x)
=0
x→a g(x)
lim
Note: the same result holds if we replace x → a with x → a+ , x → a− , x → ∞ or
x → −∞.
#
= 0”. Be careful here, we are not treating ∞ as
±∞
a real number, but merely writing something which is shorthand for the correct
statement involving limits.
Shorthand mnemonic: “
Fact (Horizontal Asymptotes of Rational Functions).
an xn + . . .
lim
x→∞ bm xm + . . .
Rule. To find


0

an
(a poly. of degree n)
=
bm
(a poly. of degree m) 


∞
if n < m
if n = m
if n > m
sum of powers of x
x→∞ sum of powers of x
we divide the top and the bottom by the combined dominant power of x that
is
� on the bottom. (“combined dominant”√ means that we take something like
2
2
�5x + 3x − 100 and simplify this is as x = x. Thus, if the bottom had
5x2 + 3x − 100 we would divide by x.)
lim