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L'Hopital's Rule ... I don't speak French, but my calc teacher in high school told me it is pronounced "low-pee-tahl" ... L'Hopital's Rule states that:

if lim f(x)=0 and lim g(x)=0 x→c x→c then limf(x)= limf'(x)x→c g(x) x→c g'(x)

For example, consider

limxx→c x^{2}+2x

This gives 0 / 0, which is indeterminate, and meets the requirements of L'Hopital's Rule. So we take the derivate of the top and bottom -- *separately*, not the derivative of the quotient. Then we find the limit of that.

limxx→c x = lim^{2}+2x2x+2x→c 1 = 2

Of course, that's a fairly boring case. We could have figured that out just by doing the division before we calculated the limit.

So let's try a more interesting problem, like

limsin xx→c x = limcos xx→c 1 = 1

For a more thorough discussion, see

© 2008 by Jay Johansen