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