Do limits evaluated at infinity exist? 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?
 A: 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 $|f(x)-L| < \epsilon$.
A: 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.
A: 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.
