5128_Ch08_434-471

454 Chapter 8 Sequences, L’Hôpital’s Rule, and Improper Integrals
which is a finite nonzero limit. The reason for this apparent disregard of common sense is
that we want “f grows faster than g” to mean that for large x-values, g is negligible in comparison
to f.
If L 1 in part 2 of the definition, then f and g are right end behavior models for each
other (Section 2.2). If f grows faster than g, then
lim f x gx
lim
x→ f x x→ ( 1 g x
f
) x
1 0 1,
so f is a right end behavior model for f g. Thus, for large x-values, g can be ignored in
the sum f g. This explains why, for large x-values, we can ignore the terms
gx a n1 x n1 … a 0
in
f x a n x n a n1 x n1 … a 0 ;
that is, why a n x n is an end behavior model for
a n x n a n1 x n1 … a 0 .
Using L’Hôpital’s Rule to Compare Growth Rates
L’Hôpital’s Rule can help us to compare rates of growth, as shown in Example 1.
EXAMPLE 1 Comparing e x and x 2 as x→
Show that the function e x grows faster than x 2 as x→.
SOLUTION
We need to show that lim x→ (e x x 2 ) . Notice this limit is of indeterminate type
, so we can apply l’Hôpital’s Rule and take the derivative of the numerator and
the derivative of the denominator. In fact, we have to apply l’Hôpital’s Rule twice.
lim e
x
x
x→ x2
lim lim
x→ 2
ex
e x
x→ 1
Now try Exercise 1.
EXPLORATION 1 Comparing Rates of Growth as x→
1. Show that a x , a 1, grows faster than x 2 as x→.
2. Show that 3 x grows faster than 2 x as x→.
3. If a b 1, show that a x grows faster than b x as x→.
EXAMPLE 2
Comparing ln x with x and x 2 as x→
Show that ln x grows slower than (a) x and (b) x 2 as x→.
SOLUTION
(a) Solve Analytically
lim ln x 1x
lim
x→ x x→ 1
l’Hôpital’s Rule
lim
x→
1 x 0
continued