# Definition of the limit as $x \to \pm\infty$ and limit is equal to $\pm\infty$

So for $$\lim_{x \to \infty}f(x)=\infty$$ by definition means:

$$\forall M>0 \exists N$$ such that $$\forall x\geq N, f(x)\geq M$$

The definition for $$\lim_{x \to -\infty}f(x)=-\infty$$ kind of confuses me. I saw from many sources they use different notation and I was wondering if both of them are correct; here they are:

$$\forall (-M)>0 \exists N$$ such that $$\forall x\leq N, f(x)\leq -M$$ and I was wondering if this is equivalent to:

$$\forall M<0 \exists N$$ such that $$\forall x\leq N, f(x)\leq M$$. Please tell me thanks!!

The second one makes more sense to me but I'm not sure if its correct.

To define $$\lim_{x \to -\infty}f(x)=-\infty$$ you want a statement that says something like "as $$x$$ approaches negative infinity, $$f(x)$$ is always smaller than any given negative number". Equivalently, given any arbitrarily large negative number $$M < 0$$, there exists an $$N < 0$$ such that $$f(x) \leq M$$ for all $$x \leq N$$. This is the same as the second definition you have stated.
$$\forall M>0 \; \exists N>0 \; s.t. \; \forall x\leq -N, f(x)\leq -M$$
This says that given any arbitrarily large negative number, we can always make $$x$$ large enough in the negative direction so that $$f(x) \leq -M$$. This is certainly equivalent to the second definition you have stated.
To see that the first definition doesn't work, notice that if $$-M >0$$ and $$f(x) \leq -M$$, then $$f$$ can take on any non-positive value; it doesn't necessarily have to approach negative infinity as $$x$$ does.