ad4: l’hoptital’s rule p569-575. l’hôpital’s rule: if has indeterminate form or, then:
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AD4: L’Hoptital’s Rule
P569-575
L’Hôpital’s Rule:
If has indeterminate form
or , then:
limx a
f x
g x
lim limx a x a
f x f x
g x g x
0
0
Ex) Indeterminate form 0/0
Evaluate x
e x
x
1lim
2
0
Ex) Indeterminate form
Evaluate x
xx
lnlim
Ex) Applying L’Hopital’s Rule More Than Once.
Evaluate xx e
x
2
lim
,0,,1,0 00
,0
0
Indeterminate forms
can be manipulated into forms so that L’Hopital’s Rule can be applied.
Determinate forms cannot be manipulated so that L’Hopital’s Rule can be applied.
0
00
Ex) Indeterminate form
Evaluate xe x
x
lim
0
Ex) Indeterminate form
Evaluate x
x x)1
1(lim
1
Ex) Indeterminate form
Evaluate x
xx)(sinlim
0
00
Ex) Indeterminate form
Evaluate )1
1
ln
1(lim
1
xxx