Let lim stand for the Limit ,
,
,
, or
, and suppose that lim and lim are both Zero or are both . If

has a finite value or if the Limit is , then

L'Hospital's rule occasionally fails to yield useful results, as in the case of the function . Repeatedly applying the rule in this case gives expressions which oscillate and never converge,

(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.

© 1996-9

1999-05-26