1Differentiation
IA Differential Equations
1.5 L’Hopital’s rule
Theorem (L’Hopital’s Rule). Let
f
(
x
) and
g
(
x
) be differentiable at
x
0
, and
lim
x→x
0
f(x) = lim
x→x
0
g(x) = 0. Then
lim
x→x
0
f(x)
g(x)
= lim
x→x
0
f
0
(x)
g
0
(x)
.
Proof.
From the Taylor’s Theorem, we have
f
(
x
) =
f
(
x
0
) + (
x − x
0
)
f
0
(
x
0
) +
o(x −x
0
), and similarly for g(x). Thus
lim
x→x
0
f(x)
g(x)
= lim
x→x
0
f(x
0
) + (x − x
0
)f
0
(x
0
) + o(x − x
0
)
g(x
0
) + (x − x
0
)g
0
(x
0
) + o(x − x
0
)
= lim
x→x
0
f
0
(x
0
) +
o(x−x
0
)
x−x
0
g
0
(x
0
) +
o(x−x
0
)
x−x
0
= lim
x→x
0
f
0
(x)
g
0
(x)