L’Hospital’s
L
Hospital s Rule
Theorem (L’Hospital’s
(L Hospital s Rule)
Suppose that a is a real number or  or . Suppose also that the
functions f and g are differentiable and g  x  0 near a (but possibly
g  a  0). Suppose also that either limxa fx  limxa gx  0 or that
limxa gx   or that limxa gx  . Finally, suppose that
f  x
lim
li
L
xa g  x
where L is either a real number or  or . Then
lim
xa
f x
 
 L.
gx
Example 1
Use L
L’Hospital’s
Hospital s Rule to show that
lim
x1
lnx
 1.
x1
Example 2
Use L’Hospital’s Rule to show that
x
e
lim
 .
x x 2
Example 3
Use L’Hospital’s Rule to show that
lnx 
lim
 0.
x 3 x
Example 4
Use L’Hospital’s
p
Rule to show that
tanx   x
1.
lim

3
x0
x3
Example 5
Explain why L’Hospital’s rule cannot be used to evaluate the limit
sinx
lim
.
x 1  cosx
Find the value of this limit by elementary means.
Example 6
Use L
L’Hospital’s
Hospital s Rule to show that
lim x lnx  0.
x0 
Hint: First note that,
that for all x  0,
0
x lnx 
lnx
1
x
Example 7
Use L’Hospital’s Rule to show that
lim
secx  tanx  0.
 
x 2
Hint: First note that, for all x for which secx and tanx are both
defined (that is, for all x for which cosx  0), we have
sinx
1  sinx
1
secx  tanx 


.
cosx
cosx
cosx
Example 8
Use L
L’Hospital’s
Hospital s Rule to show that
lim 1  sin4x cotx  e 4 .
x0
Hi t First
Hint:
Fi t let
l t y  1  sin4x
i 4  cotx and
d then
th observe
b
th t
that
lny  cotx  ln1  sin4x 
ln1  sin4x
.
tanx
Example 9
Use L’Hospital’s
p
Rule to show that
lim x x  1.
x0