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2.8. USING DERIVATIVES TO EVALUATE LIMITS
(b) lim
x→∞
e x + x
2e x + x 2
(c) lim
x→0 +
ln(x)
1
x
(d) lim
x→
π
2
−
tan(x)
x −
π
2
(e) lim
x→∞
xe
−x
⊳
When we are considering the limit of a quotient of two functions
f (x)
g(x) that results in
an indeterminate form of
∞
∞ , in essence we are asking which function is growing faster
without bound. We say that the function g dominates the function f as x → ∞ provided
that
lim
x→∞
f (x)
g(x)
= 0,
whereas f dominates g provided that lim x→∞
f (x)
g(x) = ∞. Finally, if the value of lim x→∞
f (x)
g(x)
is finite and nonzero, we say that f and g grow at the same rate. For example, from earlier
work we know that lim x→∞
x 2
e x = 0, so e x dominates x 2 , while lim x→∞
3x 2 −4x+5
7x 2 +9x−10
=
3
7 , so
f (x) = 3x 2 − 4x + 5 and g(x) = 7x 2 + 9x − 10 grow at the same rate.
Summary
In this section, we encountered the following important ideas:
• Derivatives be used to help us evaluate indeterminate limits of the form
0
0 through
L’Hopital’s Rule, which is developed by replacing the functions in the numerator and
denominator with their tangent line approximations. In particular, if f (a) = g(a) = 0
and f and g are differentiable at a, L’Hopital’s Rule tells us that
lim
x→a
f (x)
g(x)
= lim
x→a
f ′ (x)
g ′ (x)
.
• When we write x → ∞, this means that x is increasing without bound. We thus use
∞ along with limit notation to write lim x→∞ f (x) = L, which means we can make
f (x) as close to L as we like by choosing x to be sufficiently large, and similarly
lim x→a f (x) = ∞, which means we can make f (x) as large as we like by choosing x
sufficiently close to a.
• A version of L’Hopital’s Rule also allows us to use derivatives to assist us in evaluating
indeterminate limits of the form
∞
∞ . In particular, If f and g are differentiable and both