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 lim f(x) = lim f'(x)
x→c g(x) x→c g'(x)
For example, consider
lim x2+2x x→c x
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.
lim x2+2x x→c x = lim 2x+2 x→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
lim sin x x→c x = lim cos x x→c 1 = 1
For a more thorough discussion, see
© 2008 by Jay Johansen