4.4 - Indeterminate Forms and
L’Hospital’s Rule
1
L’Hospital’s Rule
Suppose f and g are differentiable functions and g'(x) ≠
0 near a (except possibly at a). Suppose that
lim f ( x)  0 and lim g ( x)  0
or that
xa
xa
lim f ( x)   and lim g ( x)  
Then
x a
x a
f ( x)
f ( x)
lim
 lim
x a g ( x)
x a g ( x)
if the limit on the right side exists (or is ±∞).
L’Hospital’s Rule
In simpler terms, if after substituting in a,
Then
f ( x) 0

lim
 or
x a g ( x)
0

f ( x)
f ( x)
lim
 lim
x a g ( x)
x a g ( x)
if the limit on the right side exists (or is ±∞).
L’Hospital’s Rule if f(a) = f(b) = 0
f ( x)  g (a )
lim
f ( x) f (a ) x a
xa
lim


x  a g ( x )
g (a ) lim g ( x)  g (a)
xa
xa
f ( x)  g (a)
f ( x)  f (a )
x

a
 lim
 lim
xa g ( x)  g (a)
x a g ( x)  g (a )
xa
f ( x)
 lim
if f (a)  g (a)  0.
xa g ( x)
Indeterminate Forms
1.
0 / 0 or ±∞ / ±∞
Strategy: Apply L’Hospital’s Rule Directly
2.
0 · ±∞
Strategy: Apply L’Hospital’s Rule to
f
g
fg 
or fg 
1/ g
1/ f
Indeterminate Forms
3.
4.
±∞ - ±∞
Strategy: Try factoring, rationalizing, finding
common denominator, etc. to get into form 1 above.
00
∞0
1∞
Properties
or
or
eln x  x
Strategy: Use a method similar to
ln x
n ln x
e

e
logarithmic differentiation. That is, take
the natural log of both sides then compute the limit.
Remember to solve for y again at the end.
n