<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] 
This is the graph of $x(\sqrt{x^2-x}-\sqrt{x^2-1})$.It is tending to $-\infty$ as $x$ is tending to $-\infty$
After rationalising 
$L=\lim_{x\to -\infty}\frac{x(1-x)}{\sqrt{x^2}(\sqrt{1-\frac{1}{x}}+\sqrt{1-\frac{1}{x^2}})}=\lim_{x\to -\infty}\frac{x(1-x)}{|x|(\sqrt{1-\frac{1}{x}}+\sqrt{1-\frac{1}{x^2}})}$
$L=\lim_{x\to -\infty}\frac{x(1-x)}{-x(\sqrt{1-\frac{1}{x}}+\sqrt{1-\frac{1}{x^2}})}=\lim_{x\to -\infty}\frac{(1-x)}{-(\sqrt{1-\frac{1}{x}}+\sqrt{1-\frac{1}{x^2}})}=-\infty$
Because when $x$ is negative ,$|x|=-x$,not $x$
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