Simplify the limand in the following way.
[tex]\displaystyle \lim_{x\to\infty} \left(\sqrt{x-a} - \sqrt{bx}\right) = \lim_{x\to\infty} \dfrac{\left(\sqrt{x-a}\right)^2 - \left(\sqrt{bx}\right)^2}{\sqrt{x-a} + \sqrt{bx}} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \lim_{x\to\infty} \frac{(x-a) - bx}{\sqrt x \left(\sqrt{1-\frac ax} + \sqrt b\right)} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \lim_{x\to\infty} \frac{(1-b)\sqrt x - \frac a{\sqrt x}}{\sqrt{1 - \frac ax} + \sqrt b}[/tex]
Now,
[tex]\displaystyle \lim_{x\to\infty} \frac a{\sqrt x} = 0[/tex]
[tex]\displaystyle \lim_{x\to\infty} \sqrt{1-\frac ax} = \sqrt{1-\lim_{x\to\infty}\frac ax}} = \sqrt1 = 1[/tex]
[tex]\implies \displaystyle \lim_{x\to\infty} \left(\sqrt{x-a} - \sqrt{bx}\right) = \frac{1-b}{\sqrt b} \lim_{x\to\infty} \sqrt x[/tex]
and therefore
[tex]\displaystyle \lim_{x\to\infty} \left(\sqrt{x-a} - \sqrt{bx}\right) = \begin{cases} 0 & \text{if } b = 1 \\ -\infty & \text{if } b > 1\end{cases}[/tex]
and does not exist otherwise.
