Fifth exercise, in the solution, how did they get -6? In my eyes, logically, the outcome'd be non-existing, due to division by 0.

This is calculus, that means the thing you should be interested in is the value of the function as x approachs 9, not when it equals 9 so the denominator will get very small but should not quite be 0.

How do they get -6 then instead of 0? Or in this case, very close to -6 instead of very close to 0?

Also, how'd they get rid of the bottom part of the division in the next to one last exercise?

OW MY BRAIN

In the next to last question, they canceled the denominator with the expanded (x-4) cubed to get (x-4)|x-4| which tends towards 0.

How does (x-4) cubed lead to 0

Found what they did in both. Seems that they did this:
9-x = (sqrt(x)+3)(-sqrt(x)+3)
And x-9 = (sqrt(x)+3)(sqrt(x)-3)
So they can scratch (sqrt(x)+3), leaving
(-sqrt(x)+3) <blablabla> / (sqrt(x)-3).
Then they put the top - outside of the brackets, giving this:
-(sqrt(x)-3) <blablabla> / (sqrt(x)-3)
You can now scratch (sqrt(x)-3), leaving:
-<blablabla>.
In the 5th exercise, it ends up beautifully in -6. In the next-to-last one, it ends up in
|x-4|?
I haven't fully figured out why they tossed one out in front, so that it looks like
(x-4)|x-4|

Thanks goes to Pogo on another forum for pointing the sqrt(x) thingy out.

Amen to that, brother. Whenever I hear this kind of discussion my eyes glaze over and I begin searching for a Klien bottle to jump into.

Meh, just a matter of not thinking "I don't understand this" on first sight, but try to follow the reasoning. That's how I'm improving on it, and I suck at math :ok:

On first sight? I've been trying to learn higher math probably for more years than you've been living, simply because I *felt* I should. I even asked my wife to explain algebra, which I took a year of back in high school long ago. Of course, she's a math head; but although she tried diligently, it didn't work. Give me words, colors, sounds; even give me the logic behind words. But I prefer the emotion behind logic. ;)