Zero over Zero

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 the limit of the fraction is the same as the limit of the fraction of the derivatives. For example,

the limit as x goes to 2 of (x^2-4)/(x^2-3x+2) *= the limit as x goes to 2 of (2x)/(2x-3).

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 2x is the derivative of x^2-4, and 2x-3 is the derivative of x^2-3x+2. We can then continue to find that

the limit as x goes to 2 of (2x)/(2x-3) = 4/1 = 4

by using the Division Limit Law.

Consider this more complicated example.

the limit as x goes to 0 of (sin(x) - x)/(x^3) *= the limit as x goes to 0 of (cos(x) - 1)/(3x^2)

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

the limit as x goes to 0 of (cos(x) - 1)/(3x^2) *= the limit as x goes to 0 of (-sin(x))/(6x) *= the limit as x goes to zero of (-cos(x))/6 = -1/6

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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