456
Chapter 4 | Applications of Derivatives
f ( x ) g ( x ) = lim x → a
f ( x )− f ( a ) g ( x )− g ( a ) since
f ( a ) =0= g ( a )
lim x → a
f ( x )− f ( a ) x − a g ( x )− g ( a ) x − a f ( x )− f ( a ) x − a g ( x )− g ( a ) x − a
= lim x → a
algebra
lim x → a lim x → a
=
limit of a quotient
f ′( a ) g ′( a )
=
definition of the derivative
f ′( x ) g ′( x )
lim x → a lim x → a
=
continuity of f ′ and g ′
f ′( x ) g ′( x ) .
= lim x → a
limit of a quotient
Note that L’Hôpital’s rule states we can calculate the limit of a quotient f
g by considering the limit of the quotient of the
f ′ g ′ . It is important to realize that we are not calculating the derivative of the quotient f g .
derivatives
□
Example 4.38 Applying L’Hôpital’s Rule (0/0 Case)
Evaluate each of the following limits by applying L’Hôpital’s rule. a. lim x →0 1−cos x x b. lim x →1 sin( πx ) ln x
e 1/ x −1 1/ x
lim x
c.
→∞
sin x − x x 2
d. lim x →0
Solution a. Since the numerator 1−cos x →0 and the denominator x →0, we can apply L’Hôpital’s rule to evaluate this limit. We have
d dx (1−cos x ) d dx ( x )
1−cos x x
lim x →0
= lim
x →0
sin x 1
= lim
x →0 lim x →0
(sin x )
=
(1)
lim x →0
= 0 1 =0.
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