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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? enter image description here

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    $\begingroup$ you will save yourself from lot of trouble, and may gain a better understanding, if only you will take time to evaluate the rational expression at, say, $x = -10.$ instead you are spending time going through someone else's work line by line. $\endgroup$ – abel Mar 8 '15 at 18:37
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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

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

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