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If $\lim\limits_{x\to a^+} f(x)=L$ and if $c$ is a function such that $a < c(x) < x$ for all $x > a$, then $\lim\limits_{x\to a^+} f(c(x))=L$. Note: there has been no discussion about continuity or any discussion about the limits of composition functions at this point. Any help would be appreciated!

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Let $\epsilon>0$. Then there exists a $\delta>0$ such that

$|f(x)-L|<\epsilon$ whenever $0. Notice that, $0, so that in particular, $0 and hence:

$|f(c(x))-L|<\epsilon$

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    I was definitel$y$ over-anal$y$zing the problem. Your explanation makes a lot of sense.Thanks2012-10-10