I was doing a homework assignment involving limits today, and something came up that didn't make any sense at all to me. In the example for finding the

**lim (x-> -Infinity) (3x-2)/sqrt(2x^2+1)**, the examples says to divide both the numerator and denominator by x. It does this in a peculiar way for the denominator though. After dividing, the resultant equation becomes:

**Now, I know the**

[ lim (x -> -Infinity) (3 - 2/x) ]/[lim (x -> -Infinity) sqrt(2x^2 + 1)/-sqrt(x^2)

[ lim (x -> -Infinity) (3 - 2/x) ]/[lim (x -> -Infinity) sqrt(2x^2 + 1)/-sqrt(x^2)

**sqrt(x^2)**is just x, but the inconsistency I see is that they divided the bottom by

**-sqrt(x^2)**instead of the positive. Furthermore, the top remains divided by positive x. This means that they're dividing the equation by x on the numerator and -x on the denominator, making it an equation divided by -1. I was under the impression that you're allowed to do whatever with 1, but not -1.

I read something about it being because x->-Infinity, which is why they use -x, but that still doesn't make sense. The numerator still needs to be divided by the same number as the denominator, so why don't they divide by -x and go with x instead?

It really frustrates me to see something that feels inconsistent like this. If anyone could shed light on why they do this, it would be much appreciated!