Difference between revisions of "L'Hospital rule"
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''l'Hôpital's rule'' | ''l'Hôpital's rule'' | ||
| − | A rule for removing indeterminacies of the form | + | 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 | + | 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 | + | 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 | + | 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 | + | (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 | + | 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) |
L'Hospital rule. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%27Hospital_rule&oldid=14236