I am currently dealing with a proof of Pauline Barrieu`s Paper "Assessing Financial Model Risk" (page 19).
At one point she applys the l'Hospital rule on a limit equation. We have some cumulative distribution function $F$ and its derivation $f$ as a density function given. I just can`t follow that step:
$$ \begin{equation} \lim_{\varepsilon \rightarrow 0}\frac{F^{-1}(\alpha)-F^{-1}(\alpha-\varepsilon)}{F^{-1}(\alpha+\varepsilon)-F^{-1}(\alpha-\varepsilon)} \end{equation} $$ after applying l'Hospital rule it should be $$ \begin{equation} \lim_{\varepsilon \rightarrow 0}\frac{1/f(\alpha-\varepsilon)}{1/f(\alpha+\varepsilon)+1/f(\alpha-\varepsilon)} \end{equation} $$
But why? I really don't get it.
My approach would be applying $[F^{-1}(\alpha)]^{\prime}=\frac{1}{f(F^{-1}(\alpha))}$
But from there I won't get any further. Also the fact that $F^{-1}(\alpha)=q_{\alpha}$ doesn't seem to help.
Maybe it's obvious, but I can't see it, so if there is someone who know how to deal with a such an equation, your help would be very much appreciated.