L'Hospital's Rule
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.
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