Do limits evaluated at infinity exist?

Limits at infinity are defined in a different way in standard analysis.

Concretely, we will say that $\lim_{x\to \infty} f(x) = L$ if for every $\epsilon >0$ there is a $M\in\mathbb{R}$ such that for every $x>M$ we have that $|f(x)-L| < \epsilon$.

The notion of limit from the left/right side simply makes no sense if $x\rightarrow\infty$ or $x\rightarrow -\infty$. The rule you're refering to is only valid when $x$ tends to a (finite) number $b$.

However limits at infinity do exist. Their definition is different from limits at $b\in\mathbb{R}$, but the same algebraic rules apply to them.

Left and right limits are special cases of the general limit concept, and only make sense for limits $x\to b$ when $b\in{\mathbb R}$. We need them in cases when a function is defined only for $x<b$, or if $f$ is given by different expressions for $x<b$ and $x>b$. The statement $$\lim_{x\to b} f(x)=\alpha\quad\Leftrightarrow\quad \lim_{x\to b-}=\alpha\quad\wedge\quad \lim_{x\to b+}=\alpha\ $$ is a proposition, and not a definition.