James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 4.4 Indeterminate Forms and l’Hospital’s Rule 305
We used a geometric argument to show that
sin x
lim − 1
x l 0 x
But these methods do not work for limits such as (1), so in this section we introduce a
sys tematic method, known as l’Hospital’s Rule, for the evaluation of indeterminate forms.
Another situation in which a limit is not obvious occurs when we look for a horizontal
asymptote of F and need to evaluate the limit
2
lim
x l `
ln x
x 2 1
It isn’t obvious how to evaluate this limit because both numerator and denominator
become large as x l `. There is a struggle between numerator and denominator. If the
numerator wins, the limit will be ` (the numerator was increasing significantly faster
than the denominator); if the denominator wins, the answer will be 0. Or there may be
some compromise, in which case the answer will be some finite positive number.
In general, if we have a limit of the form
y
f
lim
x l a
f sxd
tsxd
0
a
g
x
where both f sxd l ` (or 2`) and tsxd l ` (or 2`), then the limit may or may not
exist and is called an indeterminate form of type `y`. We saw in Section 2.6 that
this type of limit can be evaluated for certain functions, including rational functions, by
dividing numerator and denominator by the highest power of x that occurs in the denominator.
For instance,
y
y=m¡(x-a)
y=m(x-a)
lim
x l `
1 2 1 x 2 2 1
2x 2 1 1 − lim x 2
x l `
2 1 1 − 1 2 0
2 1 0 − 1 2
x 2
0
a
FIGURE 1
x
This method does not work for limits such as (2), but l’Hospital’s Rule also applies to this
type of indeterminate form.
Figure 1 suggests visually why
l’Hospital’s Rule might be true. The
first graph shows two differentiable
functions f and t, each of which
approaches 0 as x l a. If we were
to zoom in toward the point sa, 0d,
the graphs would start to look almost
linear. But if the functions actually
were linear, as in the second graph,
then their ratio would be
m 1sx 2 ad
m 2sx 2 ad − m1
m 2
which is the ratio of their derivatives.
This suggests that
fsxd
lim
x l a tsxd − lim
x l a
f9sxd
t9sxd
L’Hospital’s Rule Suppose f and t are differentiable and t9sxd ± 0 on an open
interval I that contains a (except possibly at a). Suppose that
or that
lim
x l a
f sxd − 0 and lim
x l a tsxd − 0
lim f sxd − 6` and lim tsxd − 6`
x l a x l a
(In other words, we have an indeterminate form of type 0 0 or `y`.) Then
lim
x l a
f sxd
tsxd − lim
x l a
f 9sxd
t9sxd
if the limit on the right side exists (or is ` or 2`).
Note 1 L’Hospital’s Rule says that the limit of a quotient of functions is equal to the
limit of the quotient of their derivatives, provided that the given conditions are satisfied.
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