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Section 4.5: Indeterminate Forms and L’Hospital’s Rule
Practice HW from Stewart Textbook
(not to hand in)
p. 303 # 5-39 odd
In this section, we want to be able to calculate limits
that give us indeterminate forms such as and .
In Section 2.5, we learned techniques for evaluating
these types of limit which we review in the following
examples.
0
0
Example 1: Evaluate
Solution:
3
9lim
2
3
x
xx
Example 2: Evaluate
Solution:
2
2
31
14lim
x
xx
However, the techniques of Examples 1 and 2 do not
work well if we evaluate a limit such as
For limits of this type, L’Hopital’s rule is useful.
x
e x
x
1lim
3
0
L’Hopital’s Rule
Let f and g be differentiable functions where near x = a (except possible at x = a). If
produces the indeterminate forms , , or , ,then
provided the limit exists.
0)( xg
)(
)(lim
xg
xfax
0
0
)(
)(lim
)(
)(lim
xg
xf
xg
xfaxax
Note: L’Hopital’s rule, along as the required
indeterminate form is produced, can be applied as
many times as necessary to find the limit.
Example 3: Use L’Hopital’s rule to evaluate
Solution:
3
9lim
2
3
x
xx
Example 4: Use L’Hopital’s rule to evaluate
Solution:
2
2
31
14lim
x
xx
Example 5: Evaluate
Solution:
x
e x
x
1lim
3
0
Note! We cannot apply L’Hopital’s rule if the limit
does not produce an indeterminant form , , ,
or .0
0
Example 6: Evaluate
Solution:
xx
xx
21
1lim
Helpful Fact: An expression of the form , where
, is infinite, that is, evaluates to or .0
a
0a 0
a
Example 7: Evaluate .
Solution: In typewritten notes
2
3
0
1lim
x
e x
x
Other Types of Indeterminant Forms
Note: For some functions where the limit does not
initially appear to as an indeterminant , , , or
. It may be possible to use algebraic techniques to
convert the function one of the indeterminants ,
, , or before using L’Hopital’s rule.
0
0
0
0
Indeterminant Products
Given the product of two functions , an
indeterminant of the type or results
(this is not necessarily zero!). To solve this problem,
either write the product as or and evaluate
the limit.
gf
0 0
g
f
/1 f
g
/1
Example 8: Evaluate
Solution:
x
xex
2lim
Example 9: Evaluate
Solution: In typewritten notes
xxx
ln lim 2
0
Indeterminate Differences
Get an indeterminate of the form (this is not
necessarily zero!). Usually, it is best to find a common
factor or find a common denominator to convert it into
a form where L’Hopital’s rule can be used.
Example 10: Evaluate
Solution:
xxx
cotcsclim0
Indeterminate Powers
Result in indeterminate , , or . The natural
logarithm is a useful too to write a limit of this type in
a form that L’Hopital’s rule can be used.
00 0 1
Example 11: Evaluate
Solution: (In typewritten notes)
xx
xxe
1
0)(lim