I feel like epsilon-delta is reversed The lim is about "when x approachs a, then y approachs L". Then, shouldn't the epsilon and delta be like "For all delta, no matter how small the delta is, you can always find an epsilon that makes ε < f(x)-L < ε"? But, the conventional explanation says like "for all epsilon, you find delta", which feels like to me, "when y approachs L, x goes to a".
 A: "Epsilon-Delta" is supposed to be a challenge, in order to prove that $\lim\limits_{x\to a}f(x)=L$.
Your proposal

For all delta, no matter how small the delta is, you can always find
an epsilon that makes -ε < f(x)-L < ε

Your proposal isn't much of a challenge. For any $\delta$, you can simply pick $\varepsilon = 2$, and you can "prove" that sine converges to $L = 0$ at every point.

And for any $\delta$, if you pick $\varepsilon = 10$, you can even "prove" that sine converges to $L = 5$ everywhere:

Limit definition
"For all epsilon, you find delta" describes the limit definition reasonably well.
"For all $\varepsilon$" actually means "However small $\varepsilon$ is", because large values aren't much of a challenge.

*

*For $\left\{ \begin{array}{c}
a = 0 \\ 
L = 0 \\ 
ε = 2
\end{array}
\right. 
$, can you find a corresponding $\delta$? Sure! Any delta will do, for example $\delta = \pi$:



*

*For $\left\{ \begin{array}{c}
a = 0 \\ 
L = 0 \\ 
ε = 0.5
\end{array}
\right. 
$, can you find a corresponding  $\delta$? That's a bit harder, but  $\delta = \frac{\pi}{6}$ should do:


The challenge gets progressively harder the smaller ε is. If a corresponding $\delta$ can always be found, and the curve is always completely included in the drawn rectangle $$\left\{ \begin{array}{c}
a - \delta < x < a + \delta \\ 
L - \epsilon < y < L + \epsilon
\end{array}
\right. 
$$ then $\lim\limits_{x\to a}f(x)=L$.
If you pick $\left\{ \begin{array}{c}
a = 0 \\ 
L = 0.1 \\ 
\end{array}
\right. 
$, you'll soon find an ε for which no $\delta$ fits.


If you're interested, here's the corresponding interactive diagram.
A: This is a common misunderstanding, and the only response I can ever think of is to just examine what the definition is really getting at.
The statement $\lim\limits_{x\to a}f(x)=L$ means that, when $x$ is close to $a$, $f(x)$ is close to $L$. So we want something like "if $|x-a|$ is small, then $|f(x)-L|$ is small." But then we have to decide what "small" means. If $|x-a|$ is smaller than some $\delta>0$, how small should $|f(x)-L|$ be?
The answer is that we want to ensure $|f(x)-L|$ is arbitrarily small as long as $x$ is sufficiently close to $a$. This means that, if we want $f(x)$ within $0.001$ of $L$, I can tell you how close $x$ must be to $a$. And there is no reason to find the optimal choice for this distance; the important thing is that there is some $\delta>0$ for which $|x-a|<\delta$ is enough to guarantee $|f(x)-L|<0.001$. This should work not just for $\varepsilon=0.001$, of course, but for any $\varepsilon>0$.
(By the way I should be saying $0<|x-a|<\delta$ but you get the point.)

It's also worth examining why the reverse definition is not what we want. We could try saying "for any $\delta>0$ there exists $\varepsilon>0$ such that $0<|x-a|<\delta$ implies $|f(x)-L|<\varepsilon$." The immediate issue is that $\varepsilon$ might always be huge. In the correct definition, the "for all $\varepsilon>0$" includes every arbitrarily small value for $\varepsilon$. But in the reverse definition, the smallness of $\delta$ doesn't guarantee the smallness of $\varepsilon$.
