I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia.
It says that
$f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but not equal, to the value $c$.
So this means that $f(x)$ defines $y$ or the output of the function. So when I say $f(x)$ close as desired to $L$, I actually mean the result of the calculation that has taken place and produced a $y$ close to $L$ which sits on the $y$-axis?
How close is "close enough to $c$" depends on how close one wants to make $f(x)$ to $L$.
So $c$ is actually the $x$'s that I am putting into my $f$ function. So one is picking $c$'s that are $x$'s and entering them into the function, and he actually is picking those $c$'s (sorry, $x$'s) to make his result closer to $L$, which is the limit of an approaching value of $y$?
It also of course depends on which function $f$ is, and on which number $c$ is. Therefore let the positive number $\epsilon$ be how close one wishes to make $f(x)$ to $L$;
OK, so now one picks a letter $\epsilon$ which means error, and that letter is the value of "how much one needs to be close to $L$". So it is actually the $y$ value, or the result of the function again, that needs to be close of the limit which is the $y$-coordinate again?
strictly one wants the distance to be less than $\epsilon$. Further, if the positive number $\delta$ is how close one will make $x$ to $c$,
Er, this means $\delta=x$, or the value that will be entered into $f$?
and if the distance from $x$ to $c$ is less than $\delta$ (but not zero), then the distance from $f(x)$ to $L$ will be less than $\epsilon$. Therefore $\delta$ depends on $\epsilon$. The limit statement means that no matter how small $\epsilon$ is made, $\delta$ can be made small enough.
So essentially the $\epsilon$-$\delta$ definition of the limit is the corresponding $y$, $x$ definition of the function that we use to limit it around a value? Are my conclusions wrong?
I am sorry but it seams like an "Amazing Three Cup Shuffle Magic Trick" to me on how my teacher is trying to explain this to me. I always get lost to what letters mean $\epsilon$, $\delta$, $c$, $y$, and $x$, when the function has $x$ and $y$ only.