Definition: L’Hopital’s Rule is a method we can use to find a limit when we get indeterminate form. Indeterminate form means, when we plug in, we get one of the seven following expressions:
The first, zero divided by zero, and the third, infinity divided by infinity, are the two we’re going to see the most often.
L’Hopital’s Rule: Suppose is indeterminate form. Then
In other words, if we get indeterminate form, we can take the derivative of the numerator and the denominator and take the limit of the new expression.
Example: Find the limit as x approaches 0 of .
Solution: Plugging in gives us , so we can use L’Hopital’s rule. We take the derivative of the top and the bottom, and evaluate the new limit:
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