A formal proof that the function $ x \mapsto x^{2} $ is continuous at $ x = 4 $. Problem: Show $f(x)=x^2 $ is continuous at $ x = 4$. 
That is to say, find delta such that:
$ ∀ε>0$ $ ∃δ>0 $ such that $ |x-a|<δ ⇒ |f(x)-f(a)|<ε$
Where $a=4$, $f(x)=x^2$,and $f(a)=16$. 
So in order to do to do these delta/epsilon proofs, I was originally taught to solve |f(x)-f(a)|<ε by setting it as: 
$-ε < f(x) - f(a) < ε$
Which, in this case would be: 
$-ε < x^2-16 < ε$
$16-ε < x^2 < 16+ε$
$\sqrt{16-ε} < x< \sqrt{16+ε}$
Now, since we're looking for |x-a|<δ, set $δ=\sqrt{16+ε}-4$ from the equation above. 
For the proof itself:
For $ ∀ε>0$, choose $δ=\sqrt{16+ε}-4$ and then $δ>0$ exists and implies:
$|x-a|<δ$
$ |x-4|<δ= \sqrt{16+ε}-4$
$|x-4|+4<\sqrt{16+ε}$
$|x|<\sqrt{16+ε}$
$|x^2|<16+ε$
$|x^2-16|<ε$
Q.E.D. 
The problems i'm running into are mainly with the last step. In the solutions, it uses the method of splitting up $|x^2-16|<ε$ into $|(x-4)(x+4)|<ε$ and then going from there in order to get $δ=min(1,ε/9)$. I understand how their solution works but as I was taught my way a long time ago, I'm trying to figure out why mine does/doesn't work and if it doesn't if there's a way to modify it so that it does. 
Thanks for your help!

I'd take a slightly different approach. Let $f: x \mapsto x^{2}$. Then 
\begin{align*}
|f(x) - f(4)| = |x^{2} - 4^{2}| \\
= | ((x - 4) + 4)^{2} - 4^{2}| \\
= | ((x - 4)^{2} + 2 (4)(x - 4) + 4^{2} ) - 4^{2}| \\
= | (x - 4)^{2} + 8(x - 4) | \\
\leq | x - 4 |^{2} + 8 | x - 4 |
\end{align*}
Now pick $\epsilon > 0$. If you can find $\delta > 0$ such that if $|x - 4| < \delta$, then $|x - 4|^{2} < \epsilon /2$, and $8|x - 4| < \epsilon / 2$, then you're set. Of course,
\begin{align*}
|x - 4|^{2} < \epsilon / 2 \iff |x - 4| < \sqrt{ \epsilon / 2 } \\
8|x - 4| < \epsilon / 2 \iff |x - 4| < \epsilon / 16,
\end{align*}
making it more evident that you can find such a $\delta > 0$ by picking it small enough that it meets both of the above conditions, e.g. $\delta < \min \{ \sqrt {\epsilon / 2}, \epsilon / 16 \}$. Admittedly, it takes a bit more writing, but in my opinion it's more intuitive, and is overall a reasonable method to go about it for polynomials.
