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Homework Help: Limit of an Integral

  1. Jun 19, 2010 #1
    How would I go about evaluating something like this?

    [tex]
    \[
    \lim_{n\to+\infty} n \int_0^{+\infty} \dfrac{\sin\left(\dfrac{x}{n}\right)}{x(1+x^2)}\, dx
    \]
    [/tex]
     
    Last edited: Jun 19, 2010
  2. jcsd
  3. Jun 20, 2010 #2

    arildno

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    Let I(n) be the definite integral.

    Rewrite your expression as :
    [tex]\frac{I(n)}{\frac{1}{n}}[/tex]

    Note that you can use L'hopitals rule here, and that you may interchange the operation of differentiation with respect to "n" and integration with respect to "x".
     
  4. Jun 20, 2010 #3

    arildno

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    Another way might be to write:
    [tex]\sin(\frac{x}{n})=\frac{x}{n}++++[/tex]

    Inserting, and simplifying, the limit will be the same as the above.
     
  5. Jun 20, 2010 #4

    Gib Z

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    [tex]\frac{ n \sin\left(\frac{x}{n}\right)}{x(1+x^2)} \rightarrow \frac{1}{1+x^2} [/tex] uniformly, so we can bring everything inside the integral.
     
  6. Jun 20, 2010 #5
    Thanks everyone!
     
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