For which values of the real parameter the following... How should I solve this exercise:
For which values of real parameter $a$ the following equality is true:
$$\lim_{x\to 0}{1-\cos{ax}\over x^2}=\lim_{x\to \pi}{\sin{x}\over \pi-x}$$

As pointed out by Claude Leibovici. Rewrite
$$
\lim_{x\to \pi}{\sin{x}\over \pi-x}=\lim_{\pi-x\to 0}{\sin{(\pi-x)}\over \pi-x}=1.\tag1
$$
As pointed out by  Your Ad Here, you will get
$$
\begin{align}
\lim_{x\to 0}{1-\cos{ax}\over x^2}&\stackrel{\text{l'Hospital}}=\lim_{x\to 0}{a\sin{ax}\over 2x}\\
&\stackrel{\text{l'Hospital}}=\lim_{x\to 0}{a^2\cos{ax}\over 2}\\
&=\frac{a^2}2.\tag2
\end{align}
$$
$(1)$ and $(2)$ yield $\ \Large a=\large\pm\sqrt{2}$.
