It does not hold where g prime x changes sign repeatedly as it approaches infinity, and can sometimes cause problems in cases like the one where f of x equals the quantity x plus cosine of x times sine x, and g of x equals e to the sine x power times the quantity x plus cosine x times sine x.  
It does not hold where g prime x changes sign repeatedly as it approaches infinity, and can sometimes cause problems in cases like the one where f of x equals the quantity x plus cosine of x times sine x, and g of x equals e to the sine x power times the quantity x plus cosine x times sine x.  In other circumstances, it causes repeated oscillations, which make it impossible to find the limit.  For ten points, name this rule, discovered by Bernoulli and named for the author of the textbook in which it first appears, which states that the limit of f of x divided by g of x as both approach infinity equals the limit of f prime x divided by g prime x.
It does not hold where g prime x changes sign repeatedly as it approaches infinity, and can sometimes cause problems in cases like the one where f of x equals the quantity x plus cosine of x times sine x, and g of x equals e to the sine x power times the quantity x plus cosine x times sine x.  
{L'Hopital's} rule (accept {L'Hospital's rule} or {Bernoulli's rule} before his name is mentioned)