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?

Let $t=1/(2 d x)$. Then the limit of interest can be written as 
$$\lim_{t\to 0^{\pm}}\left(\sqrt{(2a d/t + d)^2 + b^2 d^2/t^2} - \sqrt{(2a d/t - d)^2 + b^2 d^2/t^2}\right)\\
\hspace{4cm}=2d\,\text{sgn}(t)\times \lim_{t\to 0^{\pm}}\frac{1}{t}\left[\sqrt{(a+t/2)^2+b^2}-\sqrt{(a-t/2)^2+b^2}\right].$$
The definition of the derivative therefore yields the limits for $t\to 0^{\pm}$ as
$$\pm 2d \frac{d}{dt}\left( \sqrt{(a+t)^2+b^2}\right)_{t=0}=\pm \frac{2ad}{\sqrt{a^2+b^2}}.$$
