# How to show that $$\displaystyle\lim_{{{p}\to{0}}}{\left({\frac{{{p}{e}^{{{2}{t}{p}}}}}{{{1}-{e}^{{{2}{t}{p}}}{\left({1}-{p}\right)}}}}\right)}^{{k}}$$

How to show that
$\underset{p\to 0}{lim}{\left(\frac{p{e}^{2tp}}{1-{e}^{2tp}\left(1-p\right)}\right)}^{k}$
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Have you tried l'Hospital's rule?
$\underset{p\to 0}{lim}\frac{p{e}^{2tp}}{1-{e}^{2tp}\left(1-p\right)}=\underset{p\to 0}{lim}\frac{{e}^{2tp}+p\left(2t\right){e}^{2tp}}{{e}^{2tp}-\left(1-p\right)\left(2t\right){e}^{2tp}}=\underset{p\to 0}{lim}\frac{1+2tp}{1-\left(1-p\right)\left(2t\right)}=\frac{1}{1-2t}$
Just use the product rule and the case k=1 then
Hope that helps,