I am struggling with finding the limit:
$$\lim_{x\to 11} \left(\frac{x}{11}\right)^{\frac{(x-13)\cdot (x-12)}{x-11}}$$
I've tried countless methods such as turning it into the form of $a^x \to e^{\ln(a^x)}$ and yet i didn't manage to solve it. Any help would be appreciated.
Doing some arithmetics as follows leads to the definition of $e$: $$\lim_{x \to 11} (1+\frac{x-11}{11})^{\frac{11}{x-11}\cdot(x-12)(x-13)\cdot\frac{1}{11}}=e^{\lim_{x \to 11}\frac{(x-12)(x-13)}{11}}=e^{\frac{2}{11}}$$
Take logarithms, use l'Hôpital:
$\begin{align*} \lim_{x \to 11} \ln \left(\frac{x}{11}\right)^{\frac{(x - 13) (x - 12)}{x - 11}} &= \lim_{x \to 11} \frac{(x - 13) (x - 12)}{x - 11} \ln \frac{x}{11} \\ &= \lim_{x \to 11} (x - 13) (x - 12) \frac{\ln x / 11}{x - 11} \\ &= 2 \lim_{x \to 11} \frac{\ln x - \ln 11}{x - 11} \\ &= 2 \lim_{x \to 11} \frac{1/x}{1} \\ &= \frac{2}{11} \end{align*}$
So:
$\begin{align*} \lim_{x \to 11} \left(\frac{x}{11}\right)^{\frac{(x - 13) (x - 12)}{x - 11}} &= e^{2/11} \end{align*}$
