Answer
$$\frac{a-b}{2}$$
Work Step by Step
Given
$$ \lim _{x \rightarrow \infty}(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}) $$
Then
\begin{aligned}
\lim _{x \rightarrow \infty}(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}) &=\lim _{x \rightarrow \infty} \frac{(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x})(\sqrt{x^{2}+a x}+\sqrt{x^{2}+b x})}{\sqrt{x^{2}+a x}+\sqrt{x^{2}+b x}} \\
&=\lim _{x \rightarrow \infty} \frac{\left(x^{2}+a x\right)-\left(x^{2}+b x\right)}{\sqrt{x^{2}+a x}+\sqrt{x^{2}+b x}}\\
&=\lim _{x \rightarrow \infty} \frac{[(a-b) x] / x}{(\sqrt{x^{2}+ax}+\sqrt{x^{2}+b x}) / \sqrt{x^{2}}} \\ &=\lim _{x \rightarrow \infty} \frac{a-b}{\sqrt{1+a / x}+\sqrt{1+b / x}}\\
&=\frac{a-b}{\sqrt{1+0}+\sqrt{1+0}}\\
&=\frac{a-b}{2}
\end{aligned}
