<s>[INST] Evaluate: $\lim_{x\to \infty} (\sqrt {x-a} - \sqrt {bx})$ Evaluate: $\lim_{x\to \infty} (\sqrt {x-a} - \sqrt {bx})$.
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$$\lim_{x\to \infty} (\sqrt {x-a} - \sqrt {bx})$$
It takes $\infty - \infty $ form when $x=\infty $.
Now,
$$=\lim_{x\to \infty } (\sqrt {x-a} - \sqrt {bx})\times \dfrac {\sqrt {x-a}+\sqrt {bx}}{\sqrt {x-a}+\sqrt {bx}}$$
$$=\lim_{x\to \infty } \dfrac {x-a-bx}{\sqrt {x-a}+\sqrt {bx}}$$
 [/INST] Take $\sqrt{x}$ out to get:
$$L=\lim_\limits{x\to\infty} \sqrt{x}\left(\sqrt{1-\frac{a}{x}}-\sqrt{b}\right)=\lim_\limits{x\to\infty} \sqrt{x}\left(1-\sqrt{b}\right).$$
If $b=1$, $L=0$.
If $b>1$, $L=-\infty$.
If $0\le b<1$, $L=+\infty$.
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