Originally Posted by Project 2501

For most situations in which you're asked to find the limit of an indeterminate form, you can use L'Hopital's rule. This allows you to take the derivative of both the numerator and the denominator of a rational function if the limit of this function is an indeterminate form; in the case you gave, after two differentiations the question becomes the limit as x approaches 0 of (30x + 16)/(36x^2 - 32). At this point it is no longer an indeterminate form, so you can substitute and obtain the limit of -1/2.

Be aware that if you run into a function which follows a nonterminating pattern of differentiation, L'Hopital's rule will fail, because the functions will never be differentiable into a form that is not indeterminate. In those cases you'll either have to substitute or perform some clever algebra.