For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$
However, if $x\to\infty$, then what does the right hand limit mean ? How can $x\to\infty^+$, if $\infty$ is the biggest possible thing? And does this imply that all limits at $x\to\infty$ exist?
On
Even if one compactfies $\mathbb{R}$, i.e. join $\mathbb{R}$ with $+\infty$ and $-\infty$, then there is no such thing as the biggest number in standard analysis.
If you recall that, in defining one-sided limit one takes a point $c \in \mathbb{R}$ at first, then everything is clear.
Also, the phrase "as $x \to \infty$" simply means "as $x$ grows" or "as $x$ goes beyond every bound", and the phrase "as $x \to +\infty$" means "as $x$ grows" when the "direction" is unclear within context.
You should be aware that $\lim_{x\to c} f(x)$ for a finite $c$ and $\lim_{x\to\infty}f(x)$ are different concepts with separate definitions. Their notation is similar as a help to remembering them, but formally they are quite separate. In particular, $\infty$ is not "the biggest possible thing" -- it is not a "thing" at all, just a shape of ink we write instead of a thing as an informal reminder that $\lim_{x\to\infty}$ is somewhat like $\lim_{x\to c}$.
One-sided limits make sense only in the $\lim_{x\to c}$ case -- there is no directly corresponding concepts for limits at infinity.
It is certainly not the case that "all limits at $x\to\infty$" exist. For example, $ \lim_{x\to\infty} \sin x $ does not exist.