L'Hospital's Rule

Let lim stand for the Limit $\lim_{x\to c}$, $\lim_{x\to c^-}$, $\lim_{x\to c^+}$, $\lim_{x\to \infty}$, or $\lim_{x\to -\infty}$, and suppose that lim $f(x)$ and lim $g(x)$ are both Zero or are both $\pm \infty$. If

$$\lim {f'(x)\over g'(x)}$$

has a finite value or if the Limit is $\pm \infty$, then

$$\lim { f(x)\over g(x)} = \lim {f'(x)\over g'(x)}.$$

L'Hospital's rule occasionally fails to yield useful results, as in the case of the function $\lim_{u\to\infty} u(u^2+1)^{-1/2}$. Repeatedly applying the rule in this case gives expressions which oscillate and never converge,

$$\lim_{u\to\infty} {u\over (u^2+1)^{1/2}} = \lim_{u\to\infty} {1\over u(u^2+1)^{-1/2}}$$

$$= \lim_{u\to\infty} {(u^2+1)^{1/2}\over u} = \lim_{u\to\infty} {u(u^2+1)^{-1/2}\over 1}$$

$$= \lim_{u\to\infty} {u\over (u^2+1)^{1/2}}.$$

(The actual Limit is 1.)

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 13, 1972.

L'Hospital, G. de L'analyse des infiniment petits pour l'intelligence des lignes courbes. 1696.