Transcript Section 4.5: Indeterminate Forms L’Hospital’s Rule and Practice HW from Stewart Textbook

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

0
that give us indeterminate forms such as 0 and
.

In Section 2.5, we learned techniques for evaluating
these types of limit which we review in the following
examples.
Example 1: Evaluate
Solution:
x2  9
lim
x 3 x  3
Example 2: Evaluate
Solution:
lim
4x 2  1
x  1  3 x 2
However, the techniques of Examples 1 and 2 do not
work well if we evaluate a limit such as
e3x  1
lim
x 0
x
For limits of this type, L’Hopital’s rule is useful.
L’Hopital’s Rule
Let f and g be differentiable functions where
near x = a (except possible at x = a). If
g ( x)  0
f ( x)
lim
xa g ( x)
produces the indeterminate forms
then
f ( x)
lim
 lim
x a g ( x )
x a
provided the limit exists.
0
0
,  ,   or ,
f ( x)
g ( x)




,
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:
x2  9
lim
x3 x  3
Example 4: Use L’Hopital’s rule to evaluate
Solution:
lim
4x 2  1
x  1  3 x 2
Example 5: Evaluate
Solution:
e3x  1
lim
x 0
x
Note! We cannot apply L’Hopital’s rule if the limit
 
0
does not produce an indeterminant form 0 ,  ,  ,

or
.

Example 6: Evaluate lim
x 1
x 1 x 2
Solution:
x
a
0
Helpful Fact: An expression of the form
, where
a
a  0 , is infinite, that is, 0 evaluates to   or  .
Example 7: Evaluate
lim
x0
e 3x  1

x
2
Solution: In typewritten notes
.
Other Types of Indeterminant Forms
Note: For some functions where the limit does not
0  
initially appear to as an indeterminant 0 ,  ,  , or

  . It may be possible to use algebraic techniques to
0
convert the function one of the indeterminants 0 ,



 ,  , or   before using L’Hopital’s rule.
Indeterminant Products
Given the product of two functions f  g , an
indeterminant of the type 0   or   0 results
(this is not necessarily zero!). To solve this problem,
f
g
either write the product as 1 / g or 1 / f and evaluate
the limit.
Example 8: Evaluate
Solution:
lim x 2 e  x
x
Example 9: Evaluate
lim x 2 ln x
x0
Solution: In typewritten notes
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 lim csc x  cot x
x0
Solution:
Indeterminate Powers
0
0
Result in indeterminate
,  0 , or 1 . 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.
Example 11: Evaluate
lim (e x 
x0
Solution: (In typewritten notes)
1
x) x