Rule of de L'Hospital

With the rule of de L'Hospital (pronounced [ lopi'tal ], also referred to as l'Hospital's or Hospital's rule or theorem from L'Hospital , sometimes written L'Hôpital or l'Hospital ) limit values of functions that can be written as the quotient of two functions converging to zero or diverging in a certain way, and can be calculated using the first derivatives of these functions. An analogous statement for sequences instead of functions is the theorem of Stolz -Cesàro.

The rule is named after Guillaume François Antoine, Marquis de L'Hospital (1661–1704). L'Hospital published it in 1696 in his book Analyze des infiniment petits pour l'intelligence des lignes courbes , the first textbook on differential calculus . He hadn't discovered it himself, but bought it from Johann I Bernoulli .

application

In many cases, de L'Hospital's rule allows the limit value of functions to be determined even if their function term has an indefinite expression such as

${\displaystyle {\frac {0}{0}},\quad 0\cdot \infty ,\quad \infty -\infty ,\quad {\frac {\infty }{\infty }},\quad 0^{0},\quad \infty ^{0},\quad 1^{\infty }}$

supplies. All applications of the rule can be traced back to the basic task of determining the limit value when its numerator and denominator terms and either both zero or both become infinite, i.e. the quotient is an indefinite expression of the type or . De L'Hospital's rule then states that if the limit value exists, it is also the limit value , where and here the first derivatives of the functions and should be. ${\displaystyle \lim _{x\to x_{0}}{\tfrac {f(x)}{g(x)}}}$ ${\displaystyle \lim _{x\to x_{0}}{f(x)}}$ ${\displaystyle \lim _{x\to x_{0}}{g(x)}}$ ${\displaystyle {\tfrac {f(x_{0})}{g(x_{0})}}}$ ${\displaystyle {\tfrac {0}{0}}}$ ${\displaystyle {\tfrac {\infty }{\infty }}}$ ${\displaystyle \lim _{x\to x_{0}}{\tfrac {f'(x)}{g'(x)}}}$ ${\displaystyle \lim _{x\to x_{0}}{\tfrac {f(x)}{g(x)}}}$ ${\displaystyle f'}$ ${\displaystyle g'}$ ${\displaystyle f}$ ${\displaystyle g}$

However, the reverse of the rule does not apply: The fact that the limit value exists does not necessarily mean that it also exists. Therefore , if the calculation of initially provides an indefinite expression again, the numerator and denominator terms must be derived again until a specific expression is finally obtained, possibly after a finite number of repetitions . ${\displaystyle \textstyle \lim {\tfrac {f(x)}{g(x)}}}$ ${\displaystyle \textstyle \lim {\tfrac {f'(x)}{g'(x)}}}$ ${\displaystyle \lim _{x\to x_{0}}{\tfrac {f'(x)}{g'(x)}}}$

If the output function supplies a different function than the one mentioned above. indefinite expressions or , e.g. B. or , it must first be reshaped in such a way that it conforms to the above. Criteria met, i.e. it appears as the quotient of two functions that both become zero or infinite at the same time: ${\displaystyle {\tfrac {0}{0}}}$ ${\displaystyle {\tfrac {\infty }{\infty }}}$ ${\displaystyle 0\cdot \infty }$ ${\displaystyle \infty -\infty }$

example 1 ${\displaystyle :\;\;0\cdot \infty }$

${\displaystyle f(x)\cdot g(x)={\frac {f(x)}{\tfrac {1}{g(x)}}}={\frac {\phi (x)}{\psi (x)}}}$

Example 2 ${\displaystyle :\;\;\infty -\infty }$

${\displaystyle f(x)-g(x)={\frac {1}{\tfrac {1}{f(x)}}}-{\frac {1}{\tfrac {1}{g(x)}}}={\frac {{\tfrac {1}{g(x)}}-{\tfrac {1}{f(x)}}}{\tfrac {1}{f(x)\cdot g(x)}}}={\frac {\phi (x)}{\psi (x)}}}$

Precise formulation

Let be a non-empty open interval and be differentiable functions that for ( goes from below towards ) both converge to 0 or both definitely diverge. ${\displaystyle I=({\tilde {x}}_{0},x_{0})}$ ${\displaystyle f,\,g\colon I\to \mathbb {R} }$ ${\displaystyle x\nearrow x_{0}}$ ${\displaystyle x}$ ${\displaystyle x_{0}}$

If holds for all as well as for converges against a value or definitely diverges, so does so . The same applies if you replace everywhere with ( goes from above against ). ${\displaystyle g'(x)\neq 0}$ ${\displaystyle x\in I}$ ${\displaystyle {\tfrac {f'(x)}{g'(x)}}}$ ${\displaystyle x\nearrow x_{0}}$ ${\displaystyle c}$ ${\displaystyle {\tfrac {f(x)}{g(x)}}}$ ${\displaystyle x\nearrow x_{0}}$ ${\displaystyle x\searrow {\tilde {x}}_{0}}$ ${\displaystyle x}$ ${\displaystyle {\tilde {x}}_{0}}$

Is a real subset of an open interval for which the specified requirements are met, so in particular applies ${\displaystyle I}$

${\displaystyle \lim _{x\to x_{0}}{\frac {f'(x)}{g'(x)}}=c~\Rightarrow ~\lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}=c}$.

The theorem also applies to improper interval limits . ${\displaystyle x_{0}=\pm \infty }$

Evidence sketch

In the case , the functions of and at the point of continuing steadily . The theorem can thus be reduced to the extended mean value theorem , according to which, under the given conditions, there is an between and for each , so that ${\displaystyle \lim _{x\to x_{0}}f(x)=\lim _{x\to x_{0}}g(x)=0}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x_{0})=g(x_{0})=0}$ ${\displaystyle x\in I}$ ${\displaystyle \xi }$ ${\displaystyle x}$ ${\displaystyle x_{0}}$

${\displaystyle {\frac {f'(\xi )}{g'(\xi )}}={\frac {f(x_{0})-f(x)}{g(x_{0})-g(x)}}={\frac {f(x)}{g(x)}}}$.

The claim follows with the border crossing . ${\displaystyle x\nearrow x_{0}}$

The sentence can be extended to the improper case through variable transformation . ${\displaystyle x\mapsto {\tfrac {1}{x-x_{0}}}}$

Clear explanation

Approximation of two functions (solid) by their tangents (dashed)

The principle of de L'Hospital's rule is based on the fact that every pair of functions that can be differentiated at one point and thus can also be approximated there by its tangent pair there , whose equations can be formulated in the most general form (with as parameters) as follows: ${\displaystyle x_{0}}$ ${\displaystyle f(x)}$ ${\displaystyle g(x)}$ ${\displaystyle x_{0}}$

${\displaystyle f_{T}(x|x_{0})=f'(x_{0})\cdot (x-x_{0})+f(x_{0})}$ and

${\displaystyle g_{T}(x|x_{0})=g'(x_{0})\cdot (x-x_{0})+g(x_{0}).}$

As a consequence, the same must then also apply to the quotient of both functions , i.e. H. this can also be approximated by the quotient : ${\displaystyle f(x)/g(x)}$ ${\displaystyle x\to x_{0}}$ ${\displaystyle f_{T}(x|x_{0})/g_{T}(x|x_{0})}$

${\displaystyle \lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}\approx {\frac {f_{T}(x|x_{0})}{g_{T}(x|x_{0})}}={\frac {f'(x_{0})\cdot (x-x_{0})+f(x_{0})}{g'(x_{0})\cdot (x-x_{0})+g(x_{0})}}}$

If the two constants and at the same time become zero in this quotient , it gradually simplifies, as shown below, to the approximation sought: ${\displaystyle f(x_{0})}$ ${\displaystyle g(x_{0})}$

${\displaystyle \lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}\approx {\frac {f_{T}(x|x_{0})}{g_{T}(x|x_{0})}}={\frac {f'(x_{0})\cdot (x-x_{0}){\xcancel {+f(x_{0})}}}{g'(x_{0})\cdot (x-x_{0}){\xcancel {+g(x_{0})}}}}={\frac {f'(x_{0}){\xcancel {\cdot (x-x_{0})}}}{g'(x_{0}){\xcancel {\cdot (x-x_{0})}}}}={\frac {f'(x_{0})}{g'(x_{0})}}}$

Provided that and become zero at the same time, their quotient can be replaced by the quotient in the same place : ${\displaystyle f(x_{0})}$ ${\displaystyle g(x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle f(x_{0})/g(x_{0})}$ ${\displaystyle f'(x_{0})/g'(x_{0})}$

${\displaystyle f(x_{0})=0\wedge g(x_{0})=0\quad \Rightarrow \quad {\frac {f(x_{0})}{g(x_{0})}}\approx {\frac {f'(x_{0})}{g'(x_{0})}}.}$

Application examples

Limit crossing for x ₀ = 0

The convergence or divergence of . To do this, use and . The following applies: ${\displaystyle \textstyle \lim _{x\to 0}{\frac {\cos(x)-1}{\tan(x)}}}$ ${\displaystyle f(x):=\cos(x)-1}$ ${\displaystyle g(x):=\tan(x)}$

${\displaystyle \lim _{x\to 0}{f(x)}=0}$and . ${\displaystyle \lim _{x\to 0}{g(x)}=0}$

If for converges or definitely diverges, de L'Hospital's rule may be applied. Now applies: ${\displaystyle {\tfrac {f'(x)}{g'(x)}}}$ ${\displaystyle x\rightarrow 0}$

${\displaystyle {\frac {f'(x)}{g'(x)}}={\frac {-\sin(x)}{\frac {1}{\cos ^{2}(x)}}}=-\sin(x)\cos ^{2}(x)\rightarrow 0}$for . ${\displaystyle x\rightarrow 0}$

Thus, the Hospital rule is applicable. With this follows the convergence of with limit value 0. ${\displaystyle \textstyle \lim _{x\to 0}{\frac {\cos(x)-1}{\tan(x)}}}$

Border crossing in infinity

The convergence or divergence of . You bet and . Both and are definitely divergent. ${\displaystyle \textstyle \lim _{x\to \infty }{\frac {\sqrt {x}}{\ln(x)}}}$ ${\displaystyle f(x):={\sqrt {x}}}$ ${\displaystyle g(x):=\ln(x)}$ ${\displaystyle \textstyle \lim _{x\to \infty }{f(x)}=\infty }$ ${\displaystyle \textstyle \lim _{x\to \infty }{g(x)}=\infty }$

If for converges or definitely diverges, de L'Hospital's rule should be applied. Now applies ${\displaystyle {\tfrac {f'(x)}{g'(x)}}}$ ${\displaystyle x\rightarrow \infty }$

${\displaystyle {\frac {f'(x)}{g'(x)}}={{\frac {1}{2{\sqrt {x}}}} \over {\frac {1}{x}}}={\frac {\sqrt {x}}{2}}\rightarrow \infty }$for , ${\displaystyle x\rightarrow \infty }$

that is, it is definitely divergent. Therefore, the Hospital rule can be applied. The definite divergence follows from it ${\displaystyle \textstyle \lim _{x\rightarrow \infty }{\frac {f'(x)}{g'(x)}}=\infty }$

${\displaystyle \lim _{x\to \infty }{\frac {\sqrt {x}}{\ln(x)}}=\infty }$.

Warning examples

Compliance with the requirements

Be and . For is the case . ${\displaystyle \ f(x):=\sin x+2x}$ ${\displaystyle \ g(x):=\cos x+2x}$ ${\displaystyle x\to \infty }$ ${\displaystyle {\frac {\infty }{\infty }}}$

However, de L'Hospital's rule cannot be applied, because it is divergent for indefinite, since there is a periodic function. Despite the failure of Hospital's rule, converges for . Because it is . ${\displaystyle {\frac {f'(x)}{g'(x)}}={\frac {\cos x+2}{-\sin x+2}}}$ ${\displaystyle x\to \infty }$ ${\displaystyle {\frac {f(x)}{g(x)}}}$ ${\displaystyle x\rightarrow \infty }$ ${\displaystyle \lim _{x\rightarrow \infty }{\frac {f(x)}{g(x)}}=\lim _{x\rightarrow \infty }\left(1+{\frac {\sin x-\cos x}{\cos x+2x}}\right)=1}$

Landau calculus

If you want to calculate the limit value and know the Taylor expansion of denominator and numerator , it is often easier to determine the limit value using the Landau calculus than to apply de L'Hospital's rule several times. ${\displaystyle x\rightarrow x_{0}}$ ${\displaystyle x_{0}}$

So true, for example for . ${\displaystyle {\frac {\sin x-x}{x(1-\cos x)}}={\frac {-{\frac {1}{6}}x^{3}+{\mathcal {O}}(x^{5})}{x({\frac {x^{2}}{2}}+{\mathcal {O}}(x^{4}))}}={\frac {-{\frac {1}{6}}+{\mathcal {O}}(x^{2})}{{\frac {1}{2}}+{\mathcal {O}}(x^{2})}}\rightarrow -{\frac {1}{3}}}$ ${\displaystyle x\rightarrow 0}$

Generalizations

The rule can also be formulated for functions with complex variables. Are and two in holomorphic functions , which have the same order of zeros at this point . Then applies ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle D}$ ${\displaystyle a\in D}$ ${\displaystyle k}$

${\displaystyle \lim _{z\to a}{\frac {f(z)}{g(z)}}={\frac {f^{(k)}(a)}{g^{(k)}(a)}}}$.

literature

• Harro Heuser: Textbook of Analysis. Part 1 . 12th edition. Teubner, Stuttgart / Leipzig, 1998.
• Eberhard Freitag and Rolf Busam: Function Theory 1 . 3rd, revised and expanded edition. Springer-Verlag , Berlin a. a. 2000, ISBN 3-540-67641-4 .
• Otto Forster : Analysis 1 . Differential and integral calculus of a variable. 11th, expanded edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00316-6 , doi : 10.1007 / 978-3-658-00317-3 .

Web links

Wikibooks: Proof of the Rules of L'Hospital  - Learning and Teaching Materials

• The rule of L'Hospital at MathWorld (English)

Individual evidence

1. ^ Page 190 in Otto Forster : Analysis 1 . Differential and integral calculus of a variable (=  basic course in mathematics ). 12th, improved edition. Springer Spectrum, Wiesbaden 2016, ISBN 978-3-658-11544-9 , doi : 10.1007 / 978-3-658-11545-6 .
2. ↑ P. 287 in: Harro Heuser : Textbook of Analysis . Part 1. 11th edition. BG Teubner, Stuttgart 1994, ISBN 3-519-42231-X .
3. ^ Page 105 in Christiane Tretter: Analysis I (=  Mathematics Compact ). Birkhäuser, Basel 2013, ISBN 978-3-0348-0348-9 , doi : 10.1007 / 978-3-0348-0349-6 .
4. ↑ Pages 442-443 in Thomas Sonar: 3000 Years Analysis . Springer, Berlin 2011, ISBN 978-3-642-17203-8 , doi : 10.1007 / 978-3-642-17204-5 .
5. ^ W. Gellert, H. Küstner, M. Hellwich, H. Kästner : Small encyclopedia of mathematics ; Leipzig 1970, pp. 408-410.