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What does it mean for limit not to exist?

By negating the definition of limit we have: $$\exists \epsilon > 0, \quad \forall \delta >0, \exists x \in I \setminus\{c\}$$ such that $0 < |x - c| < \delta$ but $|f(x) - L| \geq \epsilon$.

I understand what this means. But what does $L$ mean here? $L$ does not even exist and in fact we are using it as a part of the definition. Can someone clarify?

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  • $\begingroup$ $L$ is a point at which $f$ doesn't approach as $x\to c$ $\endgroup$ – CIJ Oct 7 '15 at 19:12
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    $\begingroup$ This does not say that the limit does not exist. It states the condition for a given L to not be the limit. If you'd want to show that the limit does not exist, you would have to do it for all L as well $\endgroup$ – Jean-Sébastien Oct 7 '15 at 19:12
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The definition of limit should start with $\exists L$. Consequently the negation should start with $\forall L$

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$L$ is your candidate for the limit of your function. This does not mean that your function $f$ does not converge, but it does mean that $L$ is not the limit it's approaching (whether $f$ does converge or not is irrelevant). But if you take into consideration EVERY real number $L$ and then negate the definition it means that $f$ diverges as there is no limit $L$.

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