From what I know, limits only exist if both side limits exist and are equal:
$${\lim_{x\to a}f(x) = L}$$ $$if$$ $${\lim_{x\to a^+}}f(x) = {\lim_{x\to a^-}f(x) = L}$$
But can this be applied to limits at infinity? In that case: $${\lim_{x\to \infty}f(x) = L}$$ $$if$$ $${\lim_{x\to +\infty} f(x)} = {\lim_{x\to -\infty}f(x) = L}$$
Is this correct or ${\lim_{x\to \infty}f(x)}$ should be taken as ${\lim_{x\to +\infty}} f(x)$ ?