Section 3.7: L’Hospital’s Rule (concluded)
Recall the formal statement of L’Hospital’s Rule:
Suppose lim
x
→
a
f
(
x
) = 0 and lim
x
→
a
g
(
x
) = 0.
Suppose furthermore that
f
and
g
are differentiable and
g
′
(
x
)
≠
0 near
a
except possibly at
a
itself.
Then lim
x
→
a
f
(
x
)/
g
(
x
) = lim
x
→
a
f
′
(
x
)/
g
′
(
x
).
The stipulation “
g
′
(
x
)
≠
0 near
a
except possibly at
a
itself” turns out to be necessary.
There are crazy examples
where
g
′
(
a
)
≠
0, with
g
′
(
x
) oscillating wildly between
positive and negative values in the vicinity of
x
=
a
, such
that the limits lim
x
→
a
f
(
x
)/
g
(
x
) and lim
x
→
a
f
′
(
x
)/
g
′
(
x
) are
“unequal” (in the sense that one exists and the other
doesn’t).
For instance, take
f
(
x
) =
x
sin(1/
x
4
) exp(–1/
x
2
) and
g
(
x
) = exp(–1/
x
2
).
Here is what
f
(
x
)/
g
(
x
) and
f
′
(
x
)/
g
′
(
x
)
look like for
x
between 0 and 0.1:
0.02
0.04
0.06
0.08
0.10

.
0 10

.
0 05
.
0 05
.
0 10
0.02
0.04
0.06
0.08
0.10

150

100

50
50
100
150
The former approaches the limit 0 as
x
goes to 0, but the
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View Full Documentlatter approaches no limit at all!
So far we’ve looked at indeterminate expressions of type
0/0, i.e., limits of the form lim
x
→
a
f
(
x
)/
g
(
x
) with
f
(
x
),
g
(
x
)
→
0 as
x
→
a
.
Another type of indeterminate expression is typified by
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 Fall '11
 Staff
 Calculus, Limit of a function, Indeterminate form, x=a, indeterminate expression, eh eh

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