Notation for limit approaching from above/below

Question

Consider the equation $$f(x)=\frac{4x+8}{x-3}$$ It is known that $$\lim_{x \to \infty} f(x) = 4$$ from above and $$\lim_{x \to -\infty} f(x) = 4$$ from below. How do you write the "from above/below" formally as part of the equation?

Answer 1

Should be like this : $$ \lim_{x \to -\infty} f(x) = 4^- \\ \lim_{x \to +\infty} f(x) = 4^+ $$

Answer 2

In general, to express that $f(x)$ is "increasing as $x$ approaches $-\infty$", you would write (assuming the limit exists) that

$f(x)$ is strictly decreasing on $(-\infty,a)$

for some appropriate number $a$. In this case, $a=3$ is the best you can do. Similarly, $f(x)$ is strictly decreasing on $(3,\infty)$, which implies that it is approaching the limit from above.

Answer 3

The typical notations for limits at zero are \begin{align} &\lim_{x \to 0-} f(x) \\ &\lim_{x \to 0+} f(x) \end{align} This represents the limits when $x$ is restricted to negative and positive values, respectively.

One-sided limits at infinity make no sense, however. There is no way to approach $\infty$ from above