L’Hopital’s Rule
L’Hopital’s Rule Part 1
If the functions f and g are each defined in an open interval containing the number c, except
0
f ( x)
is called an intermediate form at c of the type
if
0
g ( x)

lim f ( x) = 0 and lim g ( x) = 0 and of the type
if lim f ( x) =  and lim g ( x) = 
x →c
x →c
x →c
x →c

possibly at c, then the quotient
If an intermediate form exists, then lim
x →c
the process may be repeated.
Examples:
1) Find the limit:
tan x
x→0 6 x
a) lim
b)
ln x
lim
x→ x
x
x→ e x
c) lim
f ( x)
f ( x)
. If an intermediate form still exists
= lim
x
→
c
g ( x)
g ( x)
sin x − x
x→0
x2
d) lim
You try:
ln x
x→1 x2 −1
1) lim
1− cos ( 2 )
 →0 3sin
2) lim
L’Hopital’s Part 2: L’Hopital’s Rule may also be used for Intermediate Forms of the type:
0 ,  − , 00 , 10 ,  0
In order to do this, we need to rewrite one of these forms in the form
Examples:
a)
(
)

1 
x 
lim x ln x
x→0+
b) lim  x sin
x→ 

c)
lim x x
x→0+
0

or
0

1
d)
lim+ (1+ x) x
x→0
You Try:
 
x→  


 
1) lim  x  e1/ x −1 
(
2) lim sec x − tan x
x→
2
)
L’Hopital’s Rule Assignment:
Rogawski: pg 246: 13, 18, 33, 46, 48, 50