We saw the following limit laws for limits of sequences.

Exactly the same calculation rules are valid for limits of functions:

The calculation rules for limits are also valid for the one-sided limits $\lim_{x\downarrow a}$ and $\lim_{x\uparrow a}$ and also for limits at $\pm\infty$, that is, $\lim_{x\to \infty}$ and $\lim_{x\to-\infty}$.

Just like we had for limits of sequences, we use the following conventions:

$$\begin{aligned} \infty + c &\;\;``="\ \infty \\[0.25cm] \infty + \infty &\;\;``="\ \infty \\[0.25cm] c\cdot\infty &\;\;``="\;\infty\quad \text{if } c> 0 \\[0.25cm] c\cdot\infty &\;\;``="\;-\infty\quad \text{if } c < 0 \\[0.25cm] \frac{c}{\infty} &\;\;``="\ 0 \\ \end{aligned}$$ and similar conventions for $-\infty$. If we take for example $f(x)=x$ and $g(x) = 2 + \frac{1}{x}$ then $$\lim_{x\to\infty} x\left( 2 + \frac{1}{x} \right) = 2 \cdot\infty = \infty\text.$$

Note that we again do not claim anything about expressions like

$$0 \cdot\infty, \infty - \infty\text{ and } \frac{\infty}{\infty}\text.$$ These expressions are called indeterminate forms.