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.

L'Hospital's rule must sometimes be applied with some care, since it holds only in the implicitly understood case that does not change sign infinitely often in a neighborhood of . For example, consider the limit with

as . While both and approach as , the limit of the ratio is bounded inside the interval , while the limit of approaches 0 (Boas 1986).

Another similar example is the limit with

as . While both and approach 0 as , the limit of the ratio is 0, while the limit is unbounded on the real line