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L'Hospital's rule (products)

  1. Jun 16, 2015 #1
    To deal with the indeterminate form ##0⋅\pm \infty##, we write the product ##f(x)g(x)## as ##\frac{f(x)}{1/g(x)}## or ##\frac{g(x)}{1/f(x)}##, before applying L'Hospital's rule to one of these forms.
    However, on occasion, applying L'Hospital's rule to one of these forms gets us nowhere (##\lim_{x \rightarrow -\infty} x e^x## for instance), despite working for the other (equivalent) quotient.
    Is there a way to know beforehand whether ##f(x)g(x)## should be expressed as ##\frac{f(x)}{1/g(x)}## or ##\frac{g(x)}{1/f(x)}##? Or is it always based on trial and error?
     
  2. jcsd
  3. Jun 16, 2015 #2

    RUber

    User Avatar
    Homework Helper

    I think there is some intuition needed. Usually, any function that can be reduced away should be in a position to allow that to happen, as in your example x exp(x).
    By leaving the polynomial part on top, you can be sure that application of L'Hopital's rule will eventually reduce that away.
    Other than that, I think guess and check until you've done enough...then educated guess and check.
     
  4. Jun 16, 2015 #3

    Mark44

    Staff: Mentor

    You can write ##x e^{x}## as ##\frac x {e^{-x}}##. In that form you have the form ##[\frac{-\infty}{\infty}]##, so L'Hopital's Rule can be applied.
     
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