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Thread: limit help

  1. #1
    Junior Member
    Sep 2009

    limit help

    Let f be a continuous function defined on [a,b]. Find the limit as n approaches infinity of the integral of (nf) from a to (a+1)/n.
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  2. #2
    Super Member
    Apr 2009
    (a+1)/n is not always in [a,b], don't you mean a+(1/n) (for sufficiently large n)

    Edit: Assuming you mean a+(1/n) take F(x)= \int_{a} ^{x} f(t)dt then F is differentiable in (a,b) and it's derivative is continous on [a,b] ie. F has left derivative at b and and right derivative at a.  \lim_{n \rightarrow \infty } n\int_{a} ^{a+(1/n)} f = \lim_{n \rightarrow \infty } nF(a+(1/n)) = \lim_{n \rightarrow \infty } \frac{F(a+(1/n))-F(a)}{1/n} =F'(a)=f(a)
    Last edited by Jose27; Nov 9th 2009 at 07:46 PM.
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