Section 4.8: Indeterminate Forms and L’Hopital’s Rule
Definition: A limit of the form
f (x)
0
=
x→a g(x)
0
lim
f (x)
±∞
=
x→a g(x)
±∞
or
lim
is called an indeterminate form of type 0/0 or ∞/∞.
Theorem: (L’Hopital’s Rule)
Suppose that f and g are differentiable functions and g 0 (x) 6= 0. If we have an indeterminate
form of type 0/0 or ∞/∞, then
f (x)
f 0 (x)
= lim 0
.
x→a g(x)
x→a g (x)
lim
This result is also valid for one-sided limits or limits at infinity.
Note: L’Hopital’s Rule applies ONLY for the indeterminate forms 0/0 and ∞/∞.
Example: Find each limit.
(a) lim+
x→0
x
ln x
ln x
(b) lim+ √
x→0
x
1
Example: Find each limit.
ln x
x→1 x − 1
(a) lim
sin x − x
x→0
x3
(b) lim
ex
x→∞ x3
(c) lim
ln(ex + 1)
x→∞
5x
(d) lim
2
Definition: A limit of the form
lim f (x)g(x) = 0 · ∞
x→a
is called an indeterminate form of type 0 · ∞.
Example: Find each limit.
(a) lim e−x ln x
x→∞
(b) lim xex
x→−∞
(c) lim+
√
x sec x
x→0
3
Definition: A limit of the form
lim [f (x) − g(x)] = ∞ − ∞
x→a
is called an indeterminate form of type ∞ − ∞.
Example: Find each limit.
1
1
−
(a) lim
x→1
ln x x − 1
(b) lim
x→0
(c)
1
− csc x
x
lim (sec x − tan x)
x→π/2−
4
Note: Several indeterminate forms arise from a limit of the form
lim f (x)g(x) .
x→a
The indeterminate powers are 00 , ∞0 , and 1∞ .
Example: Evaluate each limit.
(a) lim+ xsin x
x→0
(b) lim (ex + x)1/x
x→∞
5
x
2
(c) lim 1 +
x→∞
x
6