$$\lim\_{x\to 3} x^2 = 9$$

Hi, I'm new to this community (and to calculus) and English isn't my first language, so I'm sorry if I'm not able to properly convey this problem I have. If I'm doing anything wrong, could someone please correct me?

So, I know there has to be a $\delta>0$ for all $\epsilon>0$ given so that $0<|x-3|<\delta \implies |x^2-9|<\epsilon$

And I also know that $|x^2-9|<\epsilon$ can be written as $|x-3||x+3|<\epsilon$

The author (Stewart) used this method before, but I'm kind of confused...

First, he used a positive constant $C$ so that $|x+3|<C$
I'd imagine that he's trying to "restrict" $|x+3|$ by restricting the interval...
$|x-3|<1 \implies 2<x<4 \implies 5<x+3<7$

Then he chose $C = 7$, which I guess comes from the fact that it's the upper bound of $x+3$, so that $|x-3||x+3| < 7|x-3| < \epsilon \implies |x-3| < \frac{\epsilon}{7}$

And that $\delta = \min(1,\frac{\epsilon}{7})$ so that if $\epsilon > 7$, it still works.

I think I understand each individual part... But, for some reason, I feel something is missing. Why is that? Even after doing most of the exercises, this still feels alien to me.
Maybe I should try reading another book as well?
I've bought Spivak book (and it might take a while for it to arrive), does it teaches the $\epsilon - \delta$ differently?

What should I do so that I get comfortable with it?

Thanks in advance!