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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".

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  • $\begingroup$ You meant "-ε < f(x)-L < ε", right? "ε < ε" is always false. $\endgroup$ Apr 21 at 8:10
  • $\begingroup$ You've got a good answer, but if it helps, the proposed condition is equivalent to "$f$ is bounded." As the initial quantifier, "for all $\delta > 0$" is harder to meet for large $\delta$ so it incentivizes "picking $\delta>0$ as large as possible." Similarly, putting "there exists an $\varepsilon>0$" second incentivizes "picking $\varepsilon$ as large as possible." $\endgroup$ Apr 21 at 12:21

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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 $\varepsilon$.

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    $\begingroup$ I've seen the reverse before in various places and I've failed to properly articulate exactly why it's not sensible. This answer is phrased very well. I think I'll steal some of the wording next time I get asked that question. $\endgroup$ Apr 21 at 2:50
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"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.

sine and -2<y<2

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

sine and -5<y<15

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 $\delta = \pi$:

enter image description here

  • 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:

enter image description here

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.

enter image description here

enter image description here

If you're interested, here's the corresponding interactive diagram.

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