<s>[INST] How to solve following limit I've been struggeling a bit with the following limit:
$\lim\limits_{x \to 0} \frac{a- \sqrt{a^2 - x^2}}{x^2}$
The solution is:
If a < 0 then -$\infty$ .
If a > 0 then $\frac{1}{2a}$
But I don't know how to get there.
Thank you.
 [/INST] $$\begin{align*}
\lim_{x\to0}\frac{a-\sqrt{a^2-x^2}}{x^2}
&= \lim_{x\to0}\frac{a-\sqrt{a^2-x^2}}{x^2}\cdot\frac{a+\sqrt{a^2-x^2}}{a+\sqrt{a^2-x^2}}\\
&= \lim_{x\to0}\frac{x^2}{x^2\left(a+\sqrt{a^2-x^2}\right)}\\
&= \lim_{x\to0}\frac{1}{a+\sqrt{a^2-x^2}}\\
&= \frac{1}{a+\sqrt{a^2}}\\
\end{align*}$$
Now the tricky part is how you simplify $\sqrt{a^2}$ base on the sign of $a$.
</s>