<s>[INST] How to solve $\lim\limits_{x \to -\infty} \left(x\left(\sqrt{x^2-x}-\sqrt{x^2-1}\right)\right)$? I have a problem with this limit, i have no idea how to compute it.
Can you explain the method and the steps used?
$$\lim\limits_{x \to -\infty} \left(x\left(\sqrt{x^2-x}-\sqrt{x^2-1}\right)\right)$$
 [/INST] $$x\left(\sqrt{x^2-x}-\sqrt{x^2-1}\right)=\frac{x\left(\left(\sqrt{x^2-x}\right)^2-\left(\sqrt{x^2-1}\right)^2\right)}{\sqrt{x^2-x}+\sqrt{x^2-1}}$$
$$=\frac{x(1-x)}{\sqrt{x^2-x}+\sqrt{x^2-1}}$$
Since we're searching for the limit as $x\to -\infty$, let $x<0$. Then:
$$=\frac{\frac{1}{-x}(x(1-x))}{\sqrt{\frac{x^2}{(-x)^2}-\frac{x}{(-x)^2}}+\sqrt{\frac{x^2}{(-x)^2}-\frac{1}{(-x)^2}}}=\frac{x-1}{\sqrt{1-\frac{1}{x}}+\sqrt{1-\frac{1}{x^2}}}\stackrel{x\to -\infty}\to -\infty$$
Because $\sqrt{1-\frac{1}{x}}\stackrel{x\to -\infty}\to 1$ and $\sqrt{1-\frac{1}{x^2}}\stackrel{x\to -\infty}\to 1$ and $x-1\stackrel{x\to -\infty}\to -\infty$.
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