L'Hospital rule

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

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

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 \eqref{*} 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)