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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?

molarmass
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Avatrin
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  • Use $\epsilon$-$\delta$ to correctly type the hyphen (not the minus sign) in $\epsilon$-$\delta$. – Zhanxiong Nov 15 '15 at 01:05
  • I do not agree. My question is more specific. I have tried to formulate how I see limits, and I am trying to correct/tie that to the formal definition of limits. – Avatrin Nov 15 '15 at 01:12

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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 $\varepsilon$).

Martin Argerami
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  • Well, I can reformulate that. Let's say we take any sequence ${x_n}$ that converges to a. Then, the sequence ${f(x_n) }$ will converge, and its limit is L. – Avatrin Nov 15 '15 at 01:18
  • But then you have the question: what does it mean, exactly, to say that a sequence converges to a limit? In order to define this precisely, you are back to epsilons (and N's, instead of deltas). – Ned Nov 15 '15 at 01:24
  • That is true. However, the definition of converging sequences have always been more intuitive to me than limits of functions. – Avatrin Nov 15 '15 at 01:42
  • @Avatrin: sequences are functions! – Martin Argerami Nov 15 '15 at 02:14