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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
    $\displaystyle (a+1)/n$ is not always in $\displaystyle [a,b]$, don't you mean $\displaystyle a+(1/n)$ (for sufficiently large $\displaystyle n$)

    Edit: Assuming you mean $\displaystyle a+(1/n)$ take $\displaystyle F(x)= \int_{a} ^{x} f(t)dt$ then $\displaystyle F$ is differentiable in $\displaystyle (a,b)$ and it's derivative is continous on $\displaystyle [a,b]$ ie. F has left derivative at $\displaystyle b$ and and right derivative at $\displaystyle a$.$\displaystyle \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 06:46 PM.
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