How would you write formal definition for indeterminate limit?

Question

The original one, I believe, should be that for $$\lim_{{x}\to{\infty}}f(x)=L$$ $\forall\epsilon>0, \exists M \in ℝ$ such that $x>M \Rightarrow |f(x)-L|<\epsilon$

But what if it is that x approaches $-\infty$. Do you change it to $x<M$?
In addition, if the L is changed to $\infty$, how would you change the $\epsilon$?

Thanks

Answer 1

We say $\lim_{{x}\to{-\infty}}f(x)=L$ if $\forall\epsilon>0, \exists M \in \mathbb{R}$ such that $x<M \Rightarrow |f(x)-L|<\epsilon$

We say $\lim_{{x}\to{\infty}}f(x)=\infty$ if $\forall M>0, \exists N \in \mathbb{R}$ such that $x>N \Rightarrow f(x)>M$