Question: Here is some limit:
$$\lim\_{x \to b} f(x)$$

We know that for a limit to exist, we must have
$$\lim\_{x \to b+} f(x) = \lim\_{x \to b-} f(x)$$

So I am confused because, when $b=+\infty$ we can only evaluate this limit from the left side and not the right side. We can't approach infinity from a higher infinity. Does this mean that limits evaluated at infinity don't exist and therefore none of the limit laws like addition apply to limits evaluated at infinity?

EDIT: So does that mean I can use the limit laws such as addition, composition, etc on limits evaluated at infinity as long as the limits tend to a finite value?

I.e. 

$$\lim\_{x \to \infty} [f(x) + g(x)] = \lim\_{x \to \infty} f(x) + \lim\_{x \to \infty} g(x)$$ as long as both of the separate limits are some finite value?

And so on, for multiplication, composition, etc?

Answer: 
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 $|f(x)-L| < \epsilon$.
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.