proof of l’Hôpital’s rule for form
This is the proof of L’Hôpital’s Rule (http://planetmath.org/LHpitalsRule) in the case of the indeterminate form . Compared to the proof for the case (http://planetmath.org/ProofOfDeLHopitalsRule), more complicated estimates are needed.
Given . there is a such that
Let and be points such that or . (That is, both and are within distance of , but is always closer.) By Cauchy’s mean value theorem, there exists some in between and (and hence ) such that
We can assume the values , , , are all non-zero when is close enough to , say, when for some . (So there is no division by zero in our equations.) This is because and were assumed to approach , so when is close enough to , they will exceed the fixed values , , and .
but is not guaranteed to approach as approaches , so we cannot just take the limit directly. However: there exists so that
whenever . Then
- 1 Michael Spivak, Calculus, 3rd ed. Publish or Perish, 1994.
|Title||proof of l’Hôpital’s rule for form|
|Date of creation||2013-03-22 15:40:15|
|Last modified on||2013-03-22 15:40:15|
|Last modified by||stevecheng (10074)|