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The definition for limx→a f(x) = L is the following:

For all real numbers epsilon > 0, there is a real number δ > 0 such that for all real numbers x if a − δ < x < a + δ and x not equal to a then L − epsilon < f(x) < L + epsilon.

Write what it means for limx→a f(x) not equal to L.

I'm trying to find the negation for what's stated above and I was wondering if this is right.

There is not a real number such that epsilon >0, for all real numbers δ > 0 , There are real numbers x if a − δ < x < a + δ or x is equal to a then L − epsilon < f(x) < L + epsilon.

I know I'm likely wrong...what's the right answer and how do you go about finding it?

Helpsun
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    We write an expression for "It is not true that $\lim_{x\to a} f(x)=L$." First note that the definition of "limit is $L$" is not what you say it is. Now a start: There exists a positive real number $\epsilon$ such that for any positive real number $\delta$ there is a real number $x$ such that $\dots$. The part after the $\dots$ says $x\ne a$ and $x$ is near $a$ and $f(x)$ is far from $L$. – André Nicolas Sep 29 '15 at 04:50
  • @AndréNicolas I read it over and I'm wondering if the 'or' I added is correct. There exists a positive real number ϵ such that for any positive real number δ there is a real number x such that x if a − δ < x < a + δ or x≠a and x is near a and f(x) is far from L. – Helpsun Sep 29 '15 at 05:02
  • Your questions will almost certainly be answered by the time I wake up. If not, I will write an answer. – André Nicolas Sep 29 '15 at 05:11
  • @AndréNicolas Yikes, I got to know this by tomorrow morning, please tell me! – Helpsun Sep 29 '15 at 05:12

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In the definition of limit, you should not have $\delta\lt x\lt a+\delta$. It is supposed to be $a-\delta\lt x\le a+\delta$.

Now we proceed to write out the definition of "It is not the case that $\lim_{x\to a} f(x)=L$."

The first thing to note is that in the definition of limit, we start by saying that for any $\epsilon \gt 0$, a certain thing happens. To negate it, we have to say that there exists an $\epsilon\gt 0$ for which the thing does not happen.

In the definition of limit, what thing happens for any given $\epsilon$? There is a $\delta\gt 0$ with certain properties. So in writing out the negation, we have to say there exists an $\epsilon \gt 0$ such that whatever $\delta\gt 0$ we pick, a certain thing will happen.

What thing? There is an $x$ such that $a-\delta\lt x\lt a+\delta$ and $x\ne a$ and $f(x)$ is far from $L$. By far from $L$ we mean that $f(x)\le L-\epsilon$ or $f(x)\ge L+\epsilon$.

We could get at the same thing by formal manipulation of quantifiers, using the fact that $\lnot\forall t\phi$ is equivalent to $\exists t(\lnot \phi)$ and $\lnot \exists t \phi$ is equivalent to $\forall t(\lnot \phi)$.

André Nicolas
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