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2.8. USING DERIVATIVES TO EVALUATE LIMITS
assumption that g ′ is continuous at x = a)
lim
x→a
f (x)
g(x)
= lim
x→a
f ′ (x)
g ′ (x)
,
provided the righthand limit exists. This form reflects the fundamental benefit of L’Hopital’s
Rule: if
f (x)
g(x) produces an indeterminate limit of form
0
0 as x → a, it is equivalent to
consider the limit of the quotient of the two functions’ derivatives,
f ′ (x)
g ′ (x) . For example, if
we consider the limit from Preview Activity 2.8,
lim
x→1
x 5 + x − 2
x 2 − 1
,
by L’Hopital’s Rule we have that
lim
x→1
x 5 + x − 2
x 2 − 1
= lim
x→1
5x 4 + 1
2x
=
6
2
= 3.
By being able to replace the numerator and denominator with their respective derivatives,
we often move from an indeterminate limit to one whose value we can easily determine.
Activity 2.22.
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→0
ln(1 + x)
x
(b) lim
x→π
cos(x)
x
(c) lim
x→1
2 ln(x)
1 − e x−1
(d) lim
x→0
sin(x) − x
cos(2x) − 1
⊳
While L’Hopital’s Rule can be applied in an entirely algebraic way, it is important to
remember that the genesis of the rule is graphical: the main idea is that the slopes of the
tangent lines to f and g at x = a determine the value of the limit of
f (x)
g(x) as x → a. We
see this in Figure 2.20, which is a modified version of Figure 2.19, where we can see from
the grid that f ′ (a) = 2 and g ′ (a) = −1, hence by L’Hopital’s Rule,
lim
x→a
f (x)
g(x)
=
f ′ (a)
g ′ (a)
=
2
−1
= −2.
Indeed, what we observe is that it’s not the fact that f and g both approach zero that