2.8. USING DERIVATIVES TO EVALUATE LIMITS
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since
1
x 2 → 0 and
1
x → 0 as x → ∞. This shows that the rational function q has a
horizontal asymptote at y =
3
7 . A similar approach can be used to determine the limit of
any rational function as x → ∞.
But how should we handle a limit such as
lim
x→∞
x 2
e x ?
Here, both x 2 → ∞ and e x → ∞, but there is not an obvious algebraic approach that
enables us to find the limit’s value. Fortunately, it turns out that L’Hopital’s Rule extends
to cases involving infinity.
L’Hopital’s Rule (∞): If f and g are differentiable and both approach zero or both
approach ±∞ as x → a (where a is allowed to be ∞) , then
lim
x→a
f (x)
g(x)
= lim
x→a
f ′ (x)
g ′ (x)
.
(To be technically correct, we need to the additional hypothesis that g ′ (x) 0 on an
open interval that contains a or in every neighborhood of infinity if a is ∞; this is almost
always met in practice.)
To evaluate lim x→∞
x 2
e x , we observe that we can apply L’Hopital’s Rule, since both
x 2 → ∞ and e x → ∞. Doing so, it follows that
lim
x→∞
x 2
e x = lim
x→∞
2x
e x .
This updated limit is still indeterminate and of the form
∞
∞ , but it is simpler since 2x has
replaced x 2 . Hence, we can apply L’Hopital’s Rule again, by which we find that
lim
x→∞
x 2
e x = lim
x→∞
2x
e x = lim
x→∞
2
e x .
Now, since 2 is constant and e x → ∞ as x → ∞, it follows that
2
e x → 0 as x → ∞, which
shows that
lim
x→∞
x 2
e x = 0.
Activity 2.24.
Evaluate each of the following limits. If you use L’Hopital’s Rule, indicate where it was
used, and be certain its hypotheses are met before you apply it.
(a) lim
x→∞
x
ln(x)