I am struggling to properly understand the $\varepsilon$-$\delta$ definition of limits.
So, $f(x)$ gets closer to $L$ as $x$ approaches $a$. That is okay. However, taking the leap from there to the $\varepsilon$-$\delta$ definition is something I have never really been able to do.
Why is the formulation we use that we can make $|f(x) - L|$ as small as we want by making $|x - a|$ sufficiently small? How is this equivalent to the first sentence in the previous paragraph?
I could understand something like if $|x - a|$ approaches zero, so does $|f(x) - L|$. Of course, this may be harder to show algebraically. However, the $\varepsilon$-$\delta$ definition is something I simply do not understand. It may even be equivalent to the first sentence in this paragraph. I feel like it must be, but how?
When you say ``$x$ approaches $a$'', who is moving? Because numbers don't move...
What you want to say is that if $x$ is close enough to $a$, then you can guarantee that $f(x)$ is close enough to $L$. And that's exactly what the definition does: you put a limit, $\varepsilon$, regarding how far you allow $f(x)$ to be from $L$, and then you find $\delta$ that guarantees that if $x$ is close enough to $a$ (by less than $\delta$) then $f(x)$ is close enough to $L$ (by less than $\varepsilon$).