Math 151
Section 4.8
Indeterminate Forms and L’Hopital’s Rule
sin x
x!0
x
lim
lim
x!4
x 2 "16
x 2 "5x + 4
2 x "1
x!0
x
lim
Indeterminate Forms
L’Hopital’s Rule Let f and g be differentiable functions and g !( x) " 0 on an open interval I that
contains a (except possibly at a).
Then if lim f ( x) = 0 and lim g ( x ) = 0
x!a
x!a
or lim f ( x) = ±" and lim g ( x) = ±"
x!a
x!a
then lim
x!a
Cases:
f ( x)
g ( x)
= lim
x!a
f "( x)
g "( x)
if the limit exists or equals ±! .
0
!
and
0
!
Example: Evaluate the following limits.
A. lim
x!4
x 2 "16
x 2 "5x + 4
Math 151
e x + e"x " 2
x!0 1"cos 2x
( )
B. lim
e x + e"x
x!0
x2
C. lim
D. lim
ln x
x!" 3
x
4 tan x
!
x! 1+ sec x
E. lim
2
F. lim
x!"
7x
4x 2 + 5
Math 151
Case: ∞ − ∞
Example: Evaluate the following limits.
A. lim" sec x " tan x
x!
!
2
B. lim+
x!0
1
1
"
e "1 x
x
C. lim+ ln x "
x!0
1
x
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Case: 0 ⋅ ∞
Example: Evaluate the following limits.
A. lim+ x 2 ln x
x!0
B. lim" (2x " !)sec x
x!
!
2
Math 151
Cases: ∞ 0, 1∞ , 00
Example: Evaluate the following limits.
A. lim+ x x
x!0
2x
#
3&
B. lim %%1+ (((
x!" %
x ('
$