1
Section 4.5
Indeterminate Forms and
l’Hospital’s Rule
• Goals
– Introduce the various types of indeterminate
forms
– Find limits of indeterminate forms using
l’Hospital’s Rule
.
Introduction
•
Suppose we want to analyze the behavior
of the function
• Since…
– the limit of the denominator as
x
→
1 is 0 , we
cannot use Limit Laws to find
– the limit of the numerator is also 0 as
x
→
1 ,
it is not even clear whether the limit exists at
all.
(
)
=
=
−
ln
near
1.
1
x
F x
x
x
(
)
→
1
lim
;
x
F x
L’Hospital’s Rule
•
Here is a way to evaluate such limits:
L’Hospital’s Rule (cont’d)
•
Some remarks:
– L’Hospital’s Rule says that the limit of a
quotient of functions
is equal to the limit of the
quotient of their derivatives
.
– Be sure to verify the required conditions
on
f
and
g
before using l’Hospital’s Rule.
– L’Hospital’s Rule is also valid for one-sided
limits
and for limits at
∞
or –
∞
.

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