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''l'Hôpital's rule''
 
''l'Hôpital's rule''
  
A rule for removing indeterminacies of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583401.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583402.png" /> by reducing the limit of the ratio of functions to the limit of the ratio of the derivatives of the functions in question. Thus, for the case when the real-valued functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583404.png" /> are defined in a punctured right neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583405.png" /> on the number axis, l'Hospital's rule has the form
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A rule for removing indeterminacies of the form $0/0$ or $\infty/\infty$ by reducing the limit of the ratio of functions to the limit of the ratio of the derivatives of the functions in question. Thus, for the case when the real-valued functions $f$ and $g$ are defined in a punctured right neighbourhood of a point $a$ on the number axis, l'Hospital's rule has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\lim_{x\downarrow a}\frac{f(x)}{g(x)}=\lim_{x\downarrow a}\frac{f'(x)}{g'(x)}.\tag{*}$$
  
Both in the case of an indeterminacy of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583407.png" />, that is, when
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Both in the case of an indeterminacy of the form $0/0$, that is, when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583408.png" /></td> </tr></table>
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$$\lim_{x\downarrow a}f(x)=\lim_{x\downarrow a}g(x)=0,$$
  
and in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l0583409.png" />, that is, when
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and in the case $\infty/\infty$, that is, when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834010.png" /></td> </tr></table>
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$$\lim_{x\downarrow a}f(x)=\lim_{x\downarrow a}g(x)=\infty,$$
  
l'Hospital's rule is valid under the conditions that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834012.png" /> are differentiable on some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834014.png" /> for all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834015.png" />, and that there is a finite or infinite limit of the ratio of the derivatives:
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l'Hospital's rule is valid under the conditions that $f$ and $g$ are differentiable on some interval $(a,b)$, $g'(x)\neq0$ for all points $x\in(a,b)$, and that there is a finite or infinite limit of the ratio of the derivatives:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834016.png" /></td> </tr></table>
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$$\lim_{x\downarrow a}\frac{f'(x)}{g'(x)}$$
  
(in the case of an indeterminacy of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834017.png" />, this limit, if it is infinite, can only be an infinity of definite sign); then the limit of the ratio of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834018.png" /> exists and (*) holds. This assertion remains true, with natural changes, for the case of a left-sided and also a two-sided limit, and also when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834019.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834020.png" />.
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(in the case of an indeterminacy of the form $\infty/\infty$, this limit, if it is infinite, can only be an infinity of definite sign); then the limit of the ratio of the functions $\lim_{x\downarrow a}f(x)/g(x)$ exists and \ref{*} holds. This assertion remains true, with natural changes, for the case of a left-sided and also a two-sided limit, and also when $x\to+\infty$ or $x\to-\infty$.
  
 
In a practical search for limits of ratios of functions by means of l'Hospital's rule one must sometimes apply it several times in succession.
 
In a practical search for limits of ratios of functions by means of l'Hospital's rule one must sometimes apply it several times in succession.
  
Under the assumptions made above, the existence of a limit of the ratio of derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834021.png" /> is a sufficient condition for the existence of a limit of the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058340/l05834022.png" /> of the functions themselves, but it is not necessary.
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Under the assumptions made above, the existence of a limit of the ratio of derivatives $f'(x)/g'(x)$ is a sufficient condition for the existence of a limit of the ratio $f(x)/g(x)$ of the functions themselves, but it is not necessary.
  
 
====References====
 
====References====

Revision as of 15:36, 2 August 2014

l'Hôpital's rule

A rule for removing indeterminacies of the form $0/0$ or $\infty/\infty$ by reducing the limit of the ratio of functions to the limit of the ratio of the derivatives of the functions in question. Thus, for the case when the real-valued functions $f$ and $g$ are defined in a punctured right neighbourhood of a point $a$ on the number axis, l'Hospital's rule has the form

$$\lim_{x\downarrow a}\frac{f(x)}{g(x)}=\lim_{x\downarrow a}\frac{f'(x)}{g'(x)}.\tag{*}$$

Both in the case of an indeterminacy of the form $0/0$, that is, when

$$\lim_{x\downarrow a}f(x)=\lim_{x\downarrow a}g(x)=0,$$

and in the case $\infty/\infty$, that is, when

$$\lim_{x\downarrow a}f(x)=\lim_{x\downarrow a}g(x)=\infty,$$

l'Hospital's rule is valid under the conditions that $f$ and $g$ are differentiable on some interval $(a,b)$, $g'(x)\neq0$ for all points $x\in(a,b)$, and that there is a finite or infinite limit of the ratio of the derivatives:

$$\lim_{x\downarrow a}\frac{f'(x)}{g'(x)}$$

(in the case of an indeterminacy of the form $\infty/\infty$, this limit, if it is infinite, can only be an infinity of definite sign); then the limit of the ratio of the functions $\lim_{x\downarrow a}f(x)/g(x)$ exists and \ref{*} holds. This assertion remains true, with natural changes, for the case of a left-sided and also a two-sided limit, and also when $x\to+\infty$ or $x\to-\infty$.

In a practical search for limits of ratios of functions by means of l'Hospital's rule one must sometimes apply it several times in succession.

Under the assumptions made above, the existence of a limit of the ratio of derivatives $f'(x)/g'(x)$ is a sufficient condition for the existence of a limit of the ratio $f(x)/g(x)$ of the functions themselves, but it is not necessary.

References

[1] G.F. l'Hospital, "Analyse des infiniment petits pour l'intellligence des lignes courbes" , Paris (1696)
[2] S.M. Nikol'skii, "A course of mathematical analysis" , 1 , MIR (1977) (Translated from Russian)


Comments

The "rule" is probably due to Johann Bernoulli, who taught the marquis de l'Hospital mathematics.

References

[a1] G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)
[a2] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108
[a3] K.R. Stromberg, "Introduction to classical real analysis" , Wadsworth (1981)
How to Cite This Entry:
L'Hospital rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%27Hospital_rule&oldid=14236
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article