Indeterminate form0)(lim

xf

ax

0)(lim

xgax

if )(

)(lim

xg

xfax

indeterminate form of type 00

1

lnlim

1 x

xx

:Example

Indeterminate form

)(lim xf

ax

)(lim xgax

if )(

)(lim

xg

xfax

indeterminate form of type

1lim

2

xx e

x:Example

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

L’HOSPITAL’S RULE

if )(

)(lim

xg

xfax

indeterminate form of type

or 0

0

)('

)('lim

)(

)(lim

xg

xf

xg

xfaxax

1

lnlim

1 x

xx

:Example

2lim

x

ex

x

:Examplex

xx cos1

sinlim

:Example

Note:l’Hospital’s Rule can’t be applied here

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

F121

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

F101

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

Indeterminate form0)(lim

xf

ax

)(lim xgax

if)()(lim xgxf

ax

indeterminate form of type 0

xxx

lnlim0

:Example :Idea Convert into

g

ffg

/1

f

gfg

/1

If you have ln keep it in top

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

Indeterminate form

)(lim xf

ax

)(lim xgax

if )()(lim xgxfax

indeterminate form of type

)tan(seclim2

xxx

:Example

:Idea

try to convert the difference into a quotient (for instance, by using a common denominator, or rationalization, or factoring out a common factor) so that we have an indeterminate form of type 0/0 or inf/inf

q

pgf

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

F122

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

Indeterminate Powers

)()(lim xg

axxf

00

:Ideathese three cases can be treated either by taking the natural logarithm:

)(ln)(ln )( )( xfxgyxfy xg

0

1

or by writing the function as an exponential:

f(x)g(x)xg exf ln)( )(

F091

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

F101

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

F083

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

F083

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule

00

0 00 0 1

:summary

Sec 4.5: Indeterminate Forms And L’Hospital’s Rule