The following statement is called also the general mean value theorem .
Let
b
>
a
{\displaystyle {}b>a}
f
,
g
:
[
a
,
b
]
⟶
R
{\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }
are
continuous
functions which are
differentiable
on
]
a
,
b
[
{\displaystyle {}]a,b[}
g
′
(
x
)
≠
0
{\displaystyle {}g'(x)\neq 0\,}
for all
x
∈
]
a
,
b
[
{\displaystyle {}x\in {]a,b[}}
g
(
b
)
≠
g
(
a
)
{\displaystyle {}g(b)\neq g(a)}
c
∈
]
a
,
b
[
{\displaystyle {}c\in {]a,b[}}
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
=
f
′
(
c
)
g
′
(
c
)
.
{\displaystyle {}{\frac {f(b)-f(a)}{g(b)-g(a)}}={\frac {f'(c)}{g'(c)}}\,.}
The statement
g
(
a
)
≠
g
(
b
)
{\displaystyle {}g(a)\neq g(b)\,}
follows from
We consider the auxiliary function
h
(
x
)
:=
f
(
x
)
−
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
g
(
x
)
.
{\displaystyle {}h(x):=f(x)-{\frac {f(b)-f(a)}{g(b)-g(a)}}g(x)\,.}
We have
h
(
a
)
=
f
(
a
)
−
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
g
(
a
)
=
f
(
a
)
g
(
b
)
−
f
(
a
)
g
(
a
)
−
f
(
b
)
g
(
a
)
+
f
(
a
)
g
(
a
)
g
(
b
)
−
g
(
a
)
=
f
(
a
)
g
(
b
)
−
f
(
b
)
g
(
a
)
g
(
b
)
−
g
(
a
)
=
f
(
b
)
g
(
b
)
−
f
(
b
)
g
(
a
)
−
f
(
b
)
g
(
b
)
+
f
(
a
)
g
(
b
)
g
(
b
)
−
g
(
a
)
=
f
(
b
)
−
f
(
b
)
−
f
(
a
)
g
(
b
)
−
g
(
a
)
g
(
b
)
=
h
(
b
)
.
{\displaystyle {}{\begin{aligned}h(a)&=f(a)-{\frac {f(b)-f(a)}{g(b)-g(a)}}g(a)\\&={\frac {f(a)g(b)-f(a)g(a)-f(b)g(a)+f(a)g(a)}{g(b)-g(a)}}\\&={\frac {f(a)g(b)-f(b)g(a)}{g(b)-g(a)}}\\&={\frac {f(b)g(b)-f(b)g(a)-f(b)g(b)+f(a)g(b)}{g(b)-g(a)}}\\&=f(b)-{\frac {f(b)-f(a)}{g(b)-g(a)}}g(b)\\&=h(b).\end{aligned}}}
Therefore,
h
(
a
)
=
h
(
b
)
{\displaystyle {}h(a)=h(b)}
yields the existence of some
c
∈
]
a
,
b
[
{\displaystyle {}c\in {]a,b[}}
h
′
(
c
)
=
0
.
{\displaystyle {}h'(c)=0\,.}
Rearranging proves the claim.
◻
{\displaystyle \Box }
g
{\displaystyle {}g}
L’Hospital (1661-1704) For the computation of the limit of a function, the following method called L'Hôpital's rule helps.
Let
I
⊆
R
{\displaystyle {}I\subseteq \mathbb {R} }
open interval ,
and let
a
∈
I
{\displaystyle {}a\in I}
f
,
g
:
I
⟶
R
{\displaystyle f,g\colon I\longrightarrow \mathbb {R} }
are
continuous functions , which are
differentiable
on
I
∖
{
a
}
{\displaystyle {}I\setminus \{a\}}
f
(
a
)
=
g
(
a
)
=
0
{\displaystyle {}f(a)=g(a)=0}
g
′
(
x
)
≠
0
{\displaystyle {}g'(x)\neq 0}
x
≠
a
{\displaystyle {}x\neq a}
limit
w
:=
lim
x
→
a
f
′
(
x
)
g
′
(
x
)
{\displaystyle {}w:=\operatorname {lim} _{x\rightarrow a}\,{\frac {f'(x)}{g'(x)}}\,}
exists. Then also the limit
lim
x
→
a
f
(
x
)
g
(
x
)
{\displaystyle \operatorname {lim} _{x\rightarrow a}\,{\frac {f(x)}{g(x)}}}
exists, and it also equals
w
{\displaystyle {}w}
Because
g
′
{\displaystyle {}g'}
g
(
a
)
=
0
{\displaystyle {}g(a)=0}
that
a
{\displaystyle {}a}
g
{\displaystyle {}g}
(
x
n
)
n
∈
N
{\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}
sequence
in
I
∖
{
a
}
{\displaystyle {}I\setminus \{a\}}
converging
to
a
{\displaystyle {}a}
For every
x
n
{\displaystyle {}x_{n}}
applied to the interval
I
n
:=
[
x
n
,
a
]
{\displaystyle {}I_{n}:=[x_{n},a]}
[
a
,
x
n
]
{\displaystyle {}[a,x_{n}]}
c
n
{\displaystyle {}c_{n}}
[1]
I
n
{\displaystyle {}I_{n}}
f
(
x
n
)
−
f
(
a
)
g
(
x
n
)
−
g
(
a
)
=
f
′
(
c
n
)
g
′
(
c
n
)
.
{\displaystyle {}{\frac {f(x_{n})-f(a)}{g(x_{n})-g(a)}}={\frac {f'(c_{n})}{g'(c_{n})}}\,.}
The sequence
(
c
n
)
n
∈
N
{\displaystyle {}{\left(c_{n}\right)}_{n\in \mathbb {N} }}
a
{\displaystyle {}a}
f
′
(
a
)
g
′
(
a
)
=
w
{\displaystyle {}{\frac {f'(a)}{g'(a)}}=w}
w
{\displaystyle {}w}
f
(
a
)
=
g
(
a
)
=
0
{\displaystyle {}f(a)=g(a)=0}
f
(
x
n
)
g
(
x
n
)
{\displaystyle {}{\frac {f(x_{n})}{g(x_{n})}}}
w
{\displaystyle {}w}
◻
{\displaystyle \Box }
↑
The
interior
of a
real interval
I
⊆
R
{\displaystyle {}I\subseteq \mathbb {R} }
is the interval without the boundaries.
![{\displaystyle f,g\colon [a,b]\longrightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/62d73c59b59bcbf77e12fcb056dfc8687ee91d08)
![{\displaystyle {}]a,b[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/270d5e0116fb92c1667fca123c454c114a042fd9)
![{\displaystyle {}x\in {]a,b[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50d5258fad15c7c5b7180a8778c8e887d86864ef)
![{\displaystyle {}c\in {]a,b[}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/893bc6fb00faaeb9f8bf828b68e80a64823b1d6d)
![{\displaystyle {}I_{n}:=[x_{n},a]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ce5eedccbc2aec2d59e547595f32577ea92202)
![{\displaystyle {}[a,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5659198ab0c538332e1e2ffe68448a8af09f2d1)
