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.
The first definition you state is slightly incorrect though, it should read
$$\forall M>0 \; \exists N>0 \; s.t. \; \forall x\leq -N, f(x)\leq -M$$
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.