<s>[INST] Limit of function $f(x) = \sqrt{(xa + d)^2 + x^2 b^2} - \sqrt{(xa - d)^2 + x^2 b^2}$ 
I tried to calculate limit when $x$ goes to infinity for the following function 
  $$f(x) = \sqrt{(xa + d)^2 + x^2 b^2} - \sqrt{(xa - d)^2 + x^2 b^2}$$
  where $a$, $b$, $d$  are some positive constants.

It's easy to see that terms before and after minus sign goes to infinity so that gives me indeterminate symbol. Is there some way to solve this problem?
 [/INST] For $x\not=0$,
$$\begin{align}\sqrt{(xa \pm d)^2 + x^2 b^2} &=|x|\sqrt{\left(a \pm \frac{d}{x}\right)^2+ b^2}\\
&=|x|\sqrt{a^2+b^2}\left(1\pm\frac{2ad}{(a^2+b^2)x}+o(1/x)\right)^{1/2}\\
&=|x|\sqrt{a^2+b^2}\left(1\pm\frac{ad}{(a^2+b^2)x}+o(1/x)\right)\\
&=|x|\sqrt{a^2+b^2}\pm\frac{ad|x|/x}{\sqrt{a^2+b^2}}+o(1)
\end{align}$$
where we used the expansion at $t=0$, $(1+t)^{1/2}=1+t/2+o(t)$.
Now it should be easy to verify that
$$\lim\limits_{x \to +\infty} f(x)  = \frac{2ad}{\sqrt{a^2+b^2}}\quad\text{and}\quad \lim\limits_{x \to -\infty} f(x)  = -\frac{2ad}{\sqrt{a^2+b^2}}.$$
</s>