# Epsilon-Delta definition of a negative infinite limit as x approaches infinity

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.

• 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. Commented Mar 23, 2020 at 1:37
• "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. Commented Mar 23, 2020 at 2:54
• It's a question in my assignment and I thought the same thing.
– Jack
Commented Mar 23, 2020 at 3:29

## 3 Answers

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 $$f(x) . And then add quantifier : universal for $$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)

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 $$\epsilon, \delta$$.

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

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 $$M=\frac1\epsilon$$.

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.