L’Hôspital’s Rule
AP Learning Objective:
(LO 1.1C) Determine Limits of Functions
Essential Knowledge:
(EK 1.1C3) Limits of the indeterminate forms

may be evaluated using L’Hôspital’s rule.

0
and
0
Content and Practice
When finding limits, there is a group of forms that sometimes require the use of
L’Hôspital’s Rule. These are called indeterminate forms. We look for limits of the
form 0/0,  /  or expressions that can easily be written in one of these forms. As
you know, zero divided by zero is undefined, but it is vital to keep in mind that for a
limit of this form, the numerator and denominator are approaching zero, not
actually equal to zero, and L’Hôspital’s allows us to determine a limiting value.
For the form
0
, L’Hôspital’s stronger form of the rule states:
0
Suppose that f (a)  g (a)  0 , that f and g are differentiable on an open interval, I,
containing a, and that g '( x)  0 on I if x  a , then
f ( x)
f '( x)
.
L im
 L im
xa g ( x)
xa g '( x)
(over)
Find the following limits.
1) Lim
ln( x  1)
2) Lim
x  ln(3  x 2 )
sin( x)  2 x
e3 x  1
x 0
f ( x)
3
0
x
1
2
1
3) Lim    [ln( x  1)  5 x]
x   x 
g ( x)
2
5
h( x)
0
8
g '( x)
4
9
f '( x)
1
3
4) Use the table above to determine the limit, lim
x1
h '( x)
7
6
f ( g ( x))
.
h( x )
Multiple Choice Practice
1 x  x 1
3
Lim
2
x 0
x
3
1)
B) 
A) -1
2)
Lim
x 
A) 0
1
9
C) 0
D)
1
3
E)
1
2
e3 x
4 x  2e 3 x
B)
1
6
C)
1
2
D)
2
3
E) 1