[Math] Find the limit, if the limit exists: $\lim_{x\to−\infty} \sqrt{ 4x^6 − x}/( x^3 + 2)$

Question

I am wondering. Since x is approaching zero, is the strategy and goal with solving rational polynomials to determine which direction x is approaching zero from? That is, from the left or the right?

Algebraically, I am having a problem understanding the attached photo. So, will point out what I don't understand in a step rather than what I think is going on.

Nonetheless, second step, first line. Where did the negative sign come from and why? Am I to assume the negative sign is due to x approaching negative infinity*? Still, I can not make the connection as to how the two are related.

Can someone care to explain?

Answer 1

Note in general $x \neq \sqrt{x^2}$ because RHS is always positive, whereas LHS can be negative if $x<0$ (for instance $-1\neq \sqrt{(-1)^2}=1$). So in the second step if $x\to -\infty$ then clearly $x^3<0$, so we have

$$ 1/x^3=-\sqrt{1/x^6} $$