How does it transform?

[Dr.Peterson]

I also found it strange that my book replaces x with 0 instead of infinity but i found that if i do not do so, my answers wouldn't match the book's

Where do they do that? I don't see it. You are probably misinterpreting something.

Also, i notice that in this particular solution, even though they asked for the limit as x approaches negative infinity, they actually didn't even divide everything by the highest power in the denominator! Doesn't that make the answer wrong?

There are multiple ways to solve a problem; using a different method does not mean you are wrong. (They themselves show two different methods!)

But they did divide everything inside the radical by the highest power in the denominator! ("Divide all by x^3") What are you saying?

(The second method uses a known fact that can be proved by doing that.)

this guy (PatrickJMT on youtube if you know him) at 0:26, says that √x² = x.

1) So for absolute clarity, this guy is wrong, and √x² does not = x and actually = |x|. Correct?

2) He also divides everything by the highest power in the denominator, which the book does not do. Is he also wrong here?

1) That limit is as x approaches POSITIVE infinity; in that context he can assume x is positive and replace the radical with x. He just didn't state the full facts.

2) Who said that's an invalid method?

 

[bbm25]

But they did divide everything inside the radical by the highest power in the denominator! ("Divide all by x^3") What are you saying?

Forgive me. I didn't mention. I was referring to the very first picture, and not the latest one. They did not do it in the first picture, and i do not understand why they didn't

 

[bbm25]

Also, i notice that in this particular solution, even though they asked for the limit as x approaches negative infinity, they actually didn't even divide everything by the highest power in the denominator! Doesn't that make the answer wrong?

Oh actually i did mention. In this post there's a hyperlink to the first picture. But anyway, yes, in this picture, they did not divide everything by the highest power in the denominator.

 

[Dr.Peterson]

Forgive me. I didn't mention. I was referring to the very first picture, and not the latest one. They did not do it in the first picture, and i do not understand why they didn't

What they did was to factor out the highest power; when they subsequently canceled x, they were in effect dividing by it.

The reason they didn't directly divide by x was that the absolute value was needed somewhere; or, rather, that the radical intervened, muddying up the signs. There are ways they could have done that division more directly, but it probably would have confused you more!

Remember, the important thing is not that you follow some particular memorized method, but that each step is justified and takes you in a useful direction. In the same way, when you travel with a memorized route, anything that goes wrong (such as an unexpected roadblock) can cause you to get lost; when you travel with a map, or with knowledge of the region, you can take convenient alternate routes or recover from adjustments, because you know your goal more fully. In math, don't just learn a route; learn the landscape!