Factor square root out of quotient

Question

How do I get from the first expression to the second?

The only reason the limits are included is because on WolframAlpha it mentioned that this was the case for large negative numbers of x:

$\lim_{x\to\ -\infty} \frac{\sqrt{(x^2 + 2)}}{x} = \lim_{x\to\ -\infty} -\sqrt{\frac{(x^2 + 2)}{x^2}}$

Answer 1

Multiply the numerator and denominator by $-1$: $$\lim_{x\to\ -\infty} \frac{\sqrt{(x^2 + 2)}}{x} = \lim_{x\to\ -\infty} \frac{-\sqrt{(x^2 + 2)}}{-x}=\lim_{x\to\ -\infty} \frac{-\sqrt{(x^2 + 2)}}{|x|}=\\ =\lim_{x\to\ -\infty} \frac{-\sqrt{(x^2 + 2)}}{\sqrt{x^2}}=\lim_{x\to\ -\infty} -\sqrt{\frac{(x^2 + 2)}{x^2}}$$

Answer 2

Notice that we have $$\sqrt{x^2}=|x|$$

when $x<0$, we have $|x|=-x$

$$x=-|x|=-\sqrt{x^2}$$