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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,

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

(The actual Limit is 1.)


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.

© 1996-9 Eric W. Weisstein