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[Question] Limit and Integration

  1. Feb 22, 2012 #1
    Suppose f(x,y) is some continuous, smooth function with the property
    [tex]\lim_{y \to 0}f(x,y)=1
    [/tex]
    and g(x) is some other continuous smooth function.
    I want to know if this is true:
    [tex]
    \lim_{y \to 0} \int_{a}^{b}g(x)f(x,y)dx =? \int_{a}^{b}g(x)dx[/tex]

    How can I show that it is or isn't true? For which case will it be true or not true?

    Thanks
     
  2. jcsd
  3. Feb 22, 2012 #2

    micromass

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  4. Feb 22, 2012 #3
    Oh this is actually easy to see if you think of the integral as a Riemann sum.
    I dont know why I always think of the solution right after I post the question.

    So this is true in general, correct?
     
  5. Feb 23, 2012 #4

    micromass

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    No, it certainly is not true in general. And I don't really know how Riemann sums help you here.

    The problem is that you want to interchange two limits, this is not always allowed.
     
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