L'Hopital's Rule - Island of Sanity

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L'Hopital's Rule

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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 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


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