L’Hopital’s Rule

L’Hopital’s Rule is used to get you out of sticky situations with indeterminate limit forms, such as a fraction with positive/negative infinity in both the numerator and denominator, or zero times infinity. If you plug in the number you’re approaching to the function for which you’re trying to find the limit and your result is one of the indeterminate forms above, you should try applying L’Hopital’s Rule.

To use it, take the derivatives of the numerator and denominator and replace the original numerator and denominator with their derivatives. Then plug in the number you’re approaching. If you still get an indeterminate form, continue using L’Hopital’s Rule until you can use substitution to get a prettier answer. See the example on the left for more detail on applying L’Hopital’s rule.

In the example on the left, 1/2 is our final answer. However, if plugging in zero had resulted in another indeterminate form, we could have applied another round of L’Hopital’s Rule, and another and another, until we were able to plug in the number we’re approaching to get an answer that was not indeterminate.