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Homework Help: Integral estimate.

  1. May 10, 2010 #1
    1. The problem statement, all variables and given/known data
    I have to prove that the solution of an ODE can be continued to a function [tex] \in \mathcal{C}^1(\mathbb{R}) [/tex]. The solution is:
    [tex]e^{-\frac{1}{x^2}} \int_{x_0}^x -\frac{2e^{\frac{1}{t^2}}}{t^2} dt [/tex]
    It is clear that this function is not defined in [tex] x=0 [/tex]. Its limit for [tex]x \rightarrow 0 [/tex] though, seems to be zero. How do I prove it?

    2. Relevant equations
    Actually prove that the limit is zero.

    3. The attempt at a solution
    Should I use the dominated convergence theorem? Can't find the right function to dominate this one...
    Last edited: May 11, 2010
  2. jcsd
  3. May 11, 2010 #2


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    Homework Helper

    have you tried power expanding the exponentials in powers of [itex]\frac{1}{x^2} [/itex]?

    may need some extra thought regarding convergence, so not sure whether it will work, but has nice from for it, so may be worth a crack...
    Last edited: May 11, 2010
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