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