MAT 1235Calculus II

Section 6.8

Indeterminate Forms and

L’Hospital Rule

http://myhome.spu.edu/lauw

Friday

Quiz: 6.7, 6.8

Homework…

WebAssign 6.8• Part 1 (Friday, 20 problems, 57 min.)• Part 2 (Monday, 6 problems, 22 min.)

Preview

You have some experience with this topic from Lab 01

Definition of Indeterminate Forms When and How to apply the L’Hospital’s

Rule to find limits

A Common Theme in calculus -Limits

Some limits cannot be evaluated by simplifications and substitutions.

1sin

lim0

x

xx

A Common Theme in calculus -Limits

Some limits cannot be evaluated by simplifications and substitutions.

In section 2.4, we use geometric arguments to prove that

1sin

lim0

x

xx

A Common Theme in calculus -Limits

A lot of applications We have this limit in Calculus 1-3. Used to prove the derivative of

1sin

lim0

x

xx

Simple Pendulum

When the angle is small, the motion can be modeled by

l

02

2

l

g

dt

d

2

2sin 0

d g

dt l

3 5 7

sin3! 5! 7!

Definition( )

lim is called an if ( )

either (a) lim ( ) 0 and lim ( ) 0

or (b) lim ( ) and lim ( )

indeterminate for

m

x a

x a x a

x a x a

f x

g x

f x g x

f x g x

Definition0

0 Type

Type

( )lim is called an indeterminate form if

( )

either (a) lim ( ) 0 and lim ( ) 0

or (b) lim ( ) and lim ( )

x a

x a x a

x a x a

f x

g x

f x g x

f x g x

L’Hospital’s Rule

). is(or exists side handright on thelimit theif

, )(

)(lim=

)(

)(lim

Then, form. ateindeterminan is )(

)(lim that Suppose

).at possibly (except near 0)(

and abledifferenti are and Suppose

xg

xf

xg

xf

xg

xf

aaag

gf

axax

ax

Remark

xaxax , ,

casesfor validalso is rule sHospital'l' The

Example 1

0

sinlimx

x

x

Example 1

Step 1: Check that is an indeterminate form x

xx

sinlim

0

sin

0,

x

x

x

Supporting steps should be done on the right hand column.

Example 1

0 0

sinsinlim limx x

xx

x x

Step 2: Apply the l’Hospital’s rule

Remark on non-standard notation

0 0

sin coslim lim

1

H

x x

x x

x

The following non-standard notation is not acceptable in this class.

Remark on non-standard notation

The following non-standard notation is not acceptable in this class.

0 0

sin coslim lim

1

H

x x

x x

x

Example 21

0

tanlimt

t

t

1tan

0,

t

t

t

11

0 0

tantanlim limt t

tt

t t

Remarks

For some problems, you may need to apply the rule more than once. But make sure you check the condition in step 1 every time you apply the rule.

Example 3

2lim

x

x

e

x

2

,

xe

x

x

22

lim limxx

x x

ee

x x

Q&A

Q: Can I apply the l’Hospital’s rule if the limit is not in quotient form?

Q&A

Q: Can I apply the l’Hospital’s rule if the limit is not in quotient form?

A: Sometimes, we may rewrite the limit into quotient form and then apply the l’Hospital’s rule.

Example 4

0 ,x

0 0lim ln limx x

x x

0lim lnx

x x

Example 5

,x

lim limx

x xxe

lim x

xxe

Example 6

0lim x

xx

Geometric Meanings

0 0

sinlim lim cosx x

xx

x

2

1 1

1lim lim 1

1x x

xx

x

2 1

1

xy

x

1y x

Geometric Meaning

0 0

sinlim lim cosx x

xx

x

2

1 1

1lim lim 1

1x x

xx

x

sin xy

x

cosy x

2 1

1

xy

x

1y x

Reminder: ∞ is not a number

2 2

1lim

1x x

2

1lim 0x x

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