This is the standard case for l'Hôpital's Rule. If you are finding a limit of a fraction, where the limits of both the numerator and the denominator are zero, then l'Hôpital's Rule says that

.

Both the top and bottom limits of the first fraction are zero, so we can
use l'Hôpital's Rule and take derivatives. Note that 2*x* is
the derivative of *x*^2-4, and 2*x*-3 is the derivative of
*x*^2-3*x*+2. We can then continue to find that

by using the Division Limit Law.

Consider this more complicated example.

As *x* goes to zero, the limits of cos *x* - 1 and 3*x*^2
are both (still) zero, so we can apply l'Hôpital's Rule again.

Note that we used l'Hôpital's Rule twice more in that last line. As long as the limits of the numerator and denominator are still both zero (or both infinte), l'Hôpital's Rule can be used.

Copyright © 1996 Department of Mathematics, Oregon State University

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