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Where can I find a good explanation of the $\epsilon - \delta$ definition of a limit. I have tried looking at my textbook and it doesn't make much sense, and I have also looked on Google as well looking for a definition. Or maybe someone can explain it on here? I really want to understand the definition of it, but I cant seem to find an explanation that makes sense to me.

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Why don't you look at a Calculus book? I think you will find much more abot the limit there. :) –  B. S. Oct 22 '12 at 7:23
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Rather than thinking about it in terms of $\epsilon$ and $\delta$, try thinking that no matter how close to a value of $y$ you want to get, by taking your $x$ value to be sufficiently close to a certain value, you can get that close to $y$ - this is saying exactly the same thing as the $\epsilon-\delta$ limit, but you should be able to draw a picture to help you see exactly what's happening. –  user123123 Oct 22 '12 at 7:38
    
See this example. –  Mhenni Benghorbal Oct 22 '12 at 7:50
    
So then is the limit definition used for proving that a limit exists at a specific L? Or can it be used to actually find the limit of some function? I would assume both, however I don't quite see how you would use it to actually find the limit. @BabakSorouh I looked in a textbook but it didn't explain it in a way that made sense to me which is why I'm asking on here. –  TheHopefulActuary Oct 22 '12 at 16:13
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up vote 3 down vote accepted

So the definition says: $$\lim_{x \to a}f(x) = L$$ means: for all $\epsilon >0$, there exists a $\delta >0$ such that $$0<|x - a| < \delta \Rightarrow |f(x) - L| < \epsilon $$

To understand this definition, you have to know about quantifiers: (for all, there exists). In other words, If for every $\epsilon$, you are able to find a suitable $\delta$, then this proves the limit of $f$ is $L$. When you see in a statement, the word (THERE EXISTS), That means you need to find, to construct and so proving its existence. In this case, you have to find a $\delta$.

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But, I think it is a bit difficult to explain limit by the definition here. Especially when the OP aren't focusing on a certain problem or example. :-) –  B. S. Oct 22 '12 at 7:27
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And this definition is also wrong, according to the most common textbooks. The condition on $x$ must be rewritten as $0<|x-a|<\delta$, otherwise many limits will not exist. –  Siminore Oct 22 '12 at 8:13
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