In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule (French: [lopital],
English: , loh-pee-TAHL) provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.
L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if
lim
x
→
c
f
(
x
)
=
lim
x
→
c
g
(
x
)
=
0
or
±
∞
,
{\textstyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}
and
g
′
(
x
)
≠
0
{\textstyle g'(x)\neq 0}
for all x in I with x ≠ c, and
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
{\textstyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
exists, then
lim
x
→
c
f
(
x
)
g
(
x
)
=
lim
x
→
c
f
′
(
x
)
g
′
(
x
)
.
{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}.}
The differentiation of the numerator and denominator often simplifies the quotient or converts it to a limit that can be evaluated directly.