I've looked in my notes and searched online for an epsilon-delta styled definition of the limit

$\lim_{x \to +\infty} f(x) = -\infty$

I've only found the definitions in the lines of

$\forall M < 0$ $\exists N > 0 : f(x) < M$ $\forall x > N$

How do I write an epsilon-delta definition of the limit above? I tried changing M to $\epsilon$ but I haven't known of an $\epsilon$ smaller than zero in real analysis.

  • $\begingroup$ You can't, really. $\epsilon$ represents how close you need to be to the limit, but you are always infinitely far away from $-\infty$. Or, more precisely, the distance to $-\infty$ is undefined since $\infty$ is not a number. $\endgroup$ Mar 23 '20 at 1:37
  • $\begingroup$ "How do I write an epsilon-delta definition of the limit above?" Why would you want to? Finding small variances in output compared to small variance in input is utterly irrelevant in this cases. So a delta-epslion proof is both impossible and irrelevant. $\endgroup$
    – fleablood
    Mar 23 '20 at 2:54
  • $\begingroup$ It's a question in my assignment and I thought the same thing. $\endgroup$
    – Jack
    Mar 23 '20 at 3:29

$$\forall M > 0, \exists \delta > 0, : f(x) < - M, \forall x > \frac1\delta$$

Remark: I purposely write $\frac1\delta$ because $\delta$ is usually a quantity that is used to indicate a small number.

Personally, I prefer sticking to the notation of $N$ and $M$. If you want, you can let $M=\frac1\epsilon$.


The key is that the independent variable becomes larger and larger so we need some stuff like $x>M$ and $f(x) $ becomes smaller and smaller so we need something like $f(x) <N$. And then add quantifier : universal for $N$ as it deals with values of function and existential for $M$ as it is related to the independent variable (the one which occurs below limit symbol).

The desired definition of $\lim\limits_{x\to\infty} f(x) =-\infty$ is as follows:

$\forall N\in\mathbb {R}, \exists M\in\mathbb{R}: \forall x\in\mathbb {R} \, x>M\implies f(x) <N$

Notice that there is no need to specify signs of $M, N$. And if someone insists on using only the symbols $\epsilon, \delta$ just replace $M, N$ in above definition by the symbols $\epsilon, \delta$.


This is an $\epsilon/\delta$ definition. But it must be adapted to deal with infinite argument and infinite limit value. More precisely, a condition like $|x-x_0|<\epsilon$ must be written $x<-M$, meaning that $x$ is "close to minus infinity" instead of close to $x_0$ (with $M$ as large as you want corresponding to $\epsilon$ as small as you want).

You might as well write $x<-\dfrac1{\epsilon}$, but this is uncommon.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.