Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 15/latex
\setcounter{section}{15}
\subtitle {Higher derivatives}
The derivative $f'$ of a differentiable function is also called the \keyword {first derivative} {} of $f$. The zeroth derivative is the function itself. Higher derivatives are defined recursively.
\inputdefinition
{ }
{
Let
\mathrelationchain
{\relationchain
{ I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an
interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
be a
function.
The function $f$ is called $n$-times \definitionword {differentiable}{,} if it is \mathl{(n-1)}{-}times differentiable, and the \mathl{(n-1)}{-}th derivative, that is \mathl{f^{(n-1)}}{,} is also
differentiable.
The derivative
\mathrelationchaindisplay
{\relationchain
{ f^{(n)} (x)
}
{ \defeq} {(f^{(n-1)})' (x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
}
The second derivative is written as \mathl{f^{\prime \prime}}{,} the third derivative as \mathl{f^{\prime \prime \prime}}{.} If a function is $n$-times differentiable, then we say that the derivatives exist up to \keyword {order} {} $n$. A function $f$ is called \keyword {infinitely often differentiable} {,} if it is $n$-times differentiable for every $n$.
A differentiable function is continuous due to
Corollary 14.6
,
but its derivative is not necessarily so. Therefore, the following concept is justified.
\inputdefinition
{ }
{
Let
\mathrelationchain
{\relationchain
{I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an
interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
be a
function.
The function $f$ is called \definitionword {continuously differentiable}{,} if $f$ is
differentiable
and its
derivative
$f'$ is
}
A function is called $n$-times \keyword {continuously differentiable} {,} if it is $n$-times differentiable, and its $n$-th derivative is continuous.
\subtitle {Extrema of functions}
We investigate now, with the help of the methods from differentiability, when a differentiable function
\mathdisp {f \colon I \longrightarrow \R} { , }
where
\mathrelationchain
{\relationchain
{I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denotes an interval, has a
\extrabracket {local} {} {}
extremum, and how the growing behavior looks like.
\inputfactproof
{Real function/Open interval/Local extrema/Differentiable/Derivative zero/Fact}
{Theorem}
{}
{
\factsituation {Let
\mathdisp {f \colon {]a,b[} \longrightarrow \R} { }
be a
function}
\factcondition {which attains in
\mathrelationchain
{\relationchain
{c
}
{ \in }{ {]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
a
local extremum,
and is
differentiable
there.}
\factconclusion {Then
\mathrelationchain
{\relationchain
{f'(c)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds.}
\factextra {}
}
{
We may assume that $f$ attains a local maximum in $c$. This means that there exists an
\mathrelationchain
{\relationchain
{ \epsilon
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
such that
\mathrelationchain
{\relationchain
{f(x)
}
{ \leq }{f(c)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{ [c - \epsilon, c + \epsilon]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Let \mathl{{ \left( s_n \right) }_{n \in \N }}{} be a sequence with
\mathrelationchain
{\relationchain
{ c- \epsilon
}
{ \leq }{s_n
}
{ < }{ c
}
{ }{
}
{ }{
}
}
{}{}{,}
tending to $c$
\extrabracket {\quotationshort{from below}{}} {} {.}
Then
\mathrelationchain
{\relationchain
{ s_n- c
}
{ < }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and so
\mathrelationchain
{\relationchain
{f(s_n) -f(c)
}
{ \leq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and therefore the difference quotient
\mathrelationchaindisplay
{\relationchain
{ \frac{ f (s_n )-f (c) }{ s_n -c }
}
{ \geq} { 0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Due to
Lemma 7.12
,
this relation carries over to the limit, which is the derivative. Hence,
\mathrelationchain
{\relationchain
{f'(c)
}
{ \geq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For another sequence \mathl{{ \left( t_n \right) }_{n \in \N }}{} with
\mathrelationchain
{\relationchain
{ c + \epsilon
}
{ \geq }{ t_n
}
{ > }{ c
}
{ }{
}
{ }{
}
}
{}{}{,}
we get
\mathrelationchaindisplay
{\relationchain
{ \frac{ f (t_n )-f (c) }{ t_n -c }
}
{ \leq} { 0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Therefore, also
\mathrelationchain
{\relationchain
{f'(c)
}
{ \leq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and thus
\mathrelationchain
{\relationchain
{f'(c)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {X_Cubed.svg} }
\end{center}
\imagetext {} }
\imagelicense { X Cubed.svg } {} {Pieter Kuiper} {Commons} {PD} {}
We remark that the vanishing of the derivative is only a necessary, but not a sufficient, criterion for the existence of an extremum. The easiest example for this phenomenon is the function
$\R \rightarrow \R
, x \mapsto x^3$,
which is strictly increasing and its derivative is zero at the zero point. We will provide a sufficient criterion in
Corollary 15.9
below, see also
Theorem 17.4
.
\subtitle {The mean value theorem}
\inputfactproof
{Real function/Theorem of Rolle/Fact}
{Theorem}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{a
}
{ < }{b
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and let
\mathdisp {f \colon [a,b] \longrightarrow \R} { }
be a
continuous
function, which is
differentiable
on \mathl{]a,b[}{,}}
\factcondition {and such that
\mathrelationchain
{\relationchain
{f(a)
}
{ = }{f(b)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\factconclusion {Then there exists some
\mathrelationchain
{\relationchain
{c
}
{ \in }{ {]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
such that
\mathrelationchaindisplay
{\relationchain
{f'(c)
}
{ =} { 0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
The statement is true if $f$ is constant. So suppose that $f$ is not constant. Then there exists some
\mathrelationchain
{\relationchain
{x
}
{ \in }{{]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
such that
\mathrelationchain
{\relationchain
{f(x)
}
{ \neq }{ f(a)
}
{ = }{ f(b)
}
{ }{
}
{ }{
}
}
{}{}{.}
Let's say that \mathl{f(x)}{} has a larger value. Due to
Theorem 11.13
,
there exists some
\mathrelationchain
{\relationchain
{c
}
{ \in }{ [a,b]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
where the function attains its
maximum.
This point is not on the border. For this $c$, we have
\mathrelationchain
{\relationchain
{f'(c)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
due to
Theorem 15.3
.
This theorem is called \keyword {Theorem of Rolle} {.}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Mvt2_italian.svg} }
\end{center}
\imagetext {The mean value theorem means that, for every secant, there exists a parallel tangent.} }
\imagelicense { Mvt2 italian.svg } {} {4C} {Commons} {CC-by-sa 3.0} {}
The following theorem is called \keyword {Mean value theorem} {.} It says that if a function describes a differentiable one-dimensional movement, then the average velocity is obtained at least once as the instantaneous velocity.
\inputfactproof
{Differentiable functions/Mean value theorem/Fact}
{Theorem}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{a
}
{ < }{b
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and let
\mathdisp {f \colon [a,b] \longrightarrow \R} { }
be a
continuous function
which is
differentiable
on \mathl{]a,b[}{.}}
\factconclusion {Then there exists some
\mathrelationchain
{\relationchain
{c}
{ \in }{{]a,b[}}
{ }{}
{ }{}
{ }{}
}
{}{}{,} such that
\mathrelationchaindisplay
{\relationchain
{ f'(c)}
{ =} { { \frac{ f(b)-f(a) }{ b-a } } }
{ } {}
{ } {}
{ } {}
}
{}{}{.}}
\factextra {}
}
{
We consider the auxiliary function
\mathdisp {g \colon [a,b] \longrightarrow \R
, x \longmapsto g(x) \defeq f(x) - { \frac{ f(b) -f(a) }{ b-a } } (x-a)} { . }
This function is also
continuous
and
differentiable
in \mathl{]a,b[}{.} Moreover, we have
\mathrelationchain
{\relationchain
{ g(a)
}
{ = }{ f(a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathrelationchaindisplay
{\relationchain
{ g(b)
}
{ =} { f(b) -(f(b)-f(a))
}
{ =} { f(a)
}
{ } {
}
{ } {
}
}
{}{}{.}
Hence, $g$ fulfills the conditions of
Theorem 15.4
,
and therefore there exists some
\mathrelationchain
{\relationchain
{c
}
{ \in }{ {]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
such that
\mathrelationchain
{\relationchain
{g'(c)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Because of the rules for derivatives, we obtain
\mathrelationchaindisplay
{\relationchain
{ f'(c)
}
{ =} { { \frac{ f(b) -f(a) }{ b-a } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
\inputfactproof
{Real function/Derivative zero/Constant/Fact}
{Corollary}
{}
{
\factsituation {Let
\mathdisp {f \colon { ]a,b[} \longrightarrow \R} { }
be a
differentiable function}
\factcondition {such that
\mathrelationchain
{\relationchain
{ f'(x)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ {]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\factconclusion {Then $f$ is constant.}
\factextra {}
}
{
If $f$ is not constant, then there exists some
\mathrelationchain
{\relationchain
{x
}
{ < }{x'
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mathrelationchain
{\relationchain
{ f(x)
}
{ \neq }{ f(x')
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then there exists, due to
the mean value theorem,
some
\mathcond {c} {}
{x <c < x'} {}
{} {} {} {,}
such that
\mathrelationchain
{\relationchain
{f'(c)
}
{ = }{ \frac{f(x') - f(x)}{x'-x}
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
}
{}{}{,}
which contradicts the assumption.
\inputfactproof
{Real function/Derivative/Monotonicity/Fact}
{Theorem}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an
open interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
be a
differentiable function.}
\factsegue {Then the following statements hold.}
\factconclusion {\enumerationthree {The function $f$ is
increasing
\extrabracket {decreasing} {} {}
on $I$, if and only if
\mathrelationchain
{\relationchain
{f'(x)
}
{ \geq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
\extrabracket {
\mathrelationchain
{\relationchain
{f'(x)
}
{ \leq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {If
\mathrelationchain
{\relationchain
{f'(x)
}
{ \geq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and $f'$ has only finitely many
zeroes,
then $f$ is
strictly increasing.
} {If
\mathrelationchain
{\relationchain
{f'(x)
}
{ \leq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and $f'$ has only finitely many zeroes, then $f$ is
strictly decreasing.}}
\factextra {}
}
{
(1). It is enough to prove the statements for increasing functions. If $f$ is increasing and
\mathrelationchain
{\relationchain
{x
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
then the
difference quotient
fulfills
\mathrelationchaindisplay
{\relationchain
{ \frac{f(x+h) -f(x) }{h}
}
{ \geq} { 0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
for every $h$ with
\mathrelationchain
{\relationchain
{x+h
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
This estimate carries over to the limit as \mathl{h \rightarrow 0}{,} and this limit is \mathl{f'(x)}{.}
Suppose now that the derivative is $\geq 0$. We assume, in order to obtain a contradiction, that there exist two points
\mathrelationchain
{\relationchain
{x
}
{ < }{x'
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in $I$ with
\mathrelationchain
{\relationchain
{f(x)
}
{ > }{f(x')
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Due to the
mean value theorem,
there exists some $c$ with
\mathrelationchain
{\relationchain
{x
}
{ < }{c
}
{ < }{x'
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathrelationchaindisplay
{\relationchain
{f'(c)
}
{ =} {\frac{f(x') - f(x)}{x'-x}
}
{ <} {0
}
{ } {
}
{ } {
}
}
{}{}{,}
which contradicts the condition.
(2). Suppose now that
\mathrelationchain
{\relationchain
{f'(x)
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds with finitely many exceptions. We assume that
\mathrelationchain
{\relationchain
{f(x)
}
{ = }{f(x')
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for two points
\mathrelationchain
{\relationchain
{x
}
{ < }{x'
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Since $f$ is increasing, due to the first part, it follows that $f$ is constant on the interval \mathl{[x,x']}{.} But then
\mathrelationchain
{\relationchain
{f'
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
on this interval, which contradicts the condition that $f'$ has only finitely many zeroes.
\inputfactproof
{Polynomial function/Function behavior from differentiability/Fact}
{Corollary}
{}
{
A real
polynomial function
\mathdisp {f \colon \R \longrightarrow \R} { }
of
degree
\mathrelationchain
{\relationchain
{ d
}
{ \geq }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
has at most \mathl{d-1}{}
local extrema,
and one can partition the real numbers into at most $d$ intervals, on which $f$ is alternatingly
strictly increasing
or
strictly decreasing.
{See Exercise 15.14 .}
\inputfactproof
{Real function/Extrema/Second derivative/Fact}
{Corollary}
{}
{
\factsituation {Let $I$ denote a
real interval,
\mathdisp {f \colon I \longrightarrow \R} { }
a twice
continuously differentiable
function,
and
\mathrelationchain
{\relationchain
{a
}
{ \in }{I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
an inner point of the interval.}
\factcondition {Suppose that
\mathrelationchain
{\relationchain
{ f'(a)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds.}
\factsegue {Then the following statements hold.}
\factconclusion {\enumerationtwo {If
\mathrelationchain
{\relationchain
{f^{\prime \prime }(a)
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds, then $f$ has an
isolated local minimum
in $a$.
} {If
\mathrelationchain
{\relationchain
{ f^{ \prime \prime}(a)
}
{ < }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds, then $f$ has an
isolated local maximum
in $a$.
}}
\factextra {}
{See Exercise 15.15 .}
We will encounter a more general statement in
Theorem 17.4
.
\subtitle {General mean value theorem and L'Hôpital's rule}
The following statement is called also the \keyword {general mean value theorem} {.}
\inputfactproof
{Differentiable function/Mean value theorem/Quotient version/Fact}
{Theorem}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{b
}
{ > }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and suppose that
\mathdisp {f,g \colon [a,b] \longrightarrow \R} { }
are
continuous
functions which are
differentiable
on \mathl{]a,b[}{} and such that
\mathrelationchaindisplay
{\relationchain
{ g'(x)
}
{ \neq} {0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{{]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\factconclusion {Then
\mathrelationchain
{\relationchain
{ g(b)
}
{ \neq }{ g(a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and there exists some
\mathrelationchain
{\relationchain
{c
}
{ \in }{{]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
such that
\mathrelationchaindisplay
{\relationchain
{ \frac{f(b)-f(a)}{g(b)-g(a)}
}
{ =} {\frac{f'(c)}{g'(c)}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
The statement
\mathrelationchaindisplay
{\relationchain
{g(a)
}
{ \neq} {g(b)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
follows from
Theorem 15.4
.
We consider the auxiliary function
\mathrelationchaindisplay
{\relationchain
{ h(x)
}
{ \defeq} { f(x)- { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
We have
\mathrelationchainalign
{\relationchainalign
{ 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) } }
}
}
{
\relationchainextensionalign
{ =} { f(b) - { \frac{ f(b)-f(a) }{ g(b)-g(a) } } g(b)
}
{ =} { h(b)
}
{ } {}
{ } {}
}
{}{.}
Therefore,
\mathrelationchain
{\relationchain
{h(a)
}
{ = }{h(b)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and
Theorem 15.4
yields the existence of some
\mathrelationchain
{\relationchain
{c
}
{ \in }{{]a,b[}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with
\mathrelationchaindisplay
{\relationchain
{h'(c)
}
{ =} { 0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Rearranging proves the claim.
From this version, one can recover the mean value theorem, by taking for $g$ the identity.
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Guillaume_de_lHopital.jpg} }
\end{center}
\imagetext { L’Hospital (1661-1704)} }
\imagelicense { Guillaume de l'Hôpital.jpg } {} {Bemoeial} {Commons} {PD} {}
For the computation of the limit of a function, the following method called \keyword {L'Hôpital's rule} {} helps.
\inputfactproof
{Hospital/Differentiable in inner interval/Fact}
{Corollary}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an
open interval,
and let
\mathrelationchain
{\relationchain
{ a
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote a point. Suppose that
\mathdisp {f,g \colon I \longrightarrow \R} { }
are
continuous functions,}
\factcondition {which are
differentiable
on \mathl{I \setminus \{ a \}}{,} fulfilling
\mathrelationchain
{\relationchain
{ f( a )
}
{ = }{ g( a )
}
{ = }{ 0
}
{ }{
}
{ }{
}
}
{}{}{,}
and with
\mathrelationchain
{\relationchain
{ g'(x)
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for
\mathrelationchain
{\relationchain
{x
}
{ \neq }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Moreover, suppose that the
limit
\mathrelationchaindisplay
{\relationchain
{w
}
{ \defeq} { \operatorname{lim}_{ x \rightarrow a } \, \frac{f'(x)}{g'(x)}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
exists.}
\factconclusion {Then also the limit
\mathdisp {\operatorname{lim}_{ x \rightarrow a } \, \frac{f(x)}{g(x)}} { }
exists, and it also equals $w$.}
\factextra {}
}
{
Because $g'$ has no zero in the interval and
\mathrelationchain
{\relationchain
{ g(a)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds, it follows, because of
Theorem 15.4
,
that $a$ is the only zero of $g$. Let \mathl{{ \left( x_n \right) }_{n \in \N }}{} denote a
sequence
in \mathl{I \setminus \{ a \}}{,}
converging
to $a$.
For every $x_n$ there exists, due to Theorem 15.10 , applied to the interval \mathcor {} {I_n \defeq [x_n, a ]} {or} {[ a ,x_n]} {,} a $c_n$
\extrabracket {in the interior\extrafootnote {The
\definitionword {interior}{}
of a
real interval
\mathrelationchain
{\relationchain
{ I
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
of $I_n$,} {} {}
fulfilling
\mathrelationchaindisplay
{\relationchain
{ \frac{f(x_n)-f( a )}{g( x_n )-g( a ) }
}
{ =} { \frac{f'(c_n)}{g'(c_n)}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The sequence \mathl{{ \left( c_n \right) }_{n \in \N }}{} converges also to $a$, so that, because of the condition, the right-hand side converges to
\mathrelationchain
{\relationchain
{ \frac{f'( a )}{g'( a )}
}
{ = }{ w
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Therefore, also the left-hand side converges to $w$, and, because of
\mathrelationchain
{\relationchain
{ f( a )
}
{ = }{ g( a )
}
{ = }{ 0
}
{ }{
}
{ }{
}
}
{}{}{,}
this means that \mathl{\frac{f(x_n)}{g(x_n)}}{} converges to $w$.
\inputexample{}
{
The
polynomials
\mathdisp {3x^2-5x-2 \text{ and } x^3-4x^2+x+6} { }
have both a zero for
\mathrelationchain
{\relationchain
{x
}
{ = }{2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
It is therefore not immediately clear whether the limit
\mathdisp {\operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6}} { }
exists. Applying twice
L'Hôpital's rule,
we get the existence and
\mathrelationchaindisplay
{\relationchain
{ \operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 3x^2-5x-2}{x^3-4x^2+x+6}
}
{ =} { \operatorname{lim}_{ x \rightarrow 2 } \, \frac{ 6x-5}{3x^2-8x+1}
}
{ =} { \frac{7}{-3}
}
{ =} { - \frac{7}{3}
}
{ } {
}
}
{}{}{.}
}