Evaluate $\lim_{x\rightarrow \infty} (\sqrt{x^2 +ax} - \sqrt{x^2 +bx})$ Evaluate $$\lim_{x\rightarrow \infty} (\sqrt{x^2 +ax} - \sqrt{x^2 +bx})$$
I tried the following:
$$\lim_{x\rightarrow \infty} (\sqrt{x^2 +ax} - \sqrt{x^2 +bx}) \cdot \frac{\sqrt{x^2 +ax} + \sqrt{x^2 +bx}}{\sqrt{x^2 +ax} + \sqrt{x^2 +bx}}$$
But ended up with
$$\lim_{x\rightarrow \infty} \frac{ax-bx}{\sqrt{x^2 +ax} + \sqrt{x^2 +bx}}$$
Which I'm not sure what to do with.

another method: asymptotics
For $x>0$, we have as $x \to \infty$:
$$
\sqrt{x^2+ax} = x\sqrt{1+\frac{a}{x}} = x\left(1+\frac{a}{2x}+o(x^{-1})\right) = x+\frac{a}{2}+o(1)
\\
\sqrt{x^2+bx}= x+\frac{b}{2}+o(1)
\\
\sqrt{x^2+ax}-\sqrt{x^2+bx}= \frac{a-b}{2}+o(1)
$$
