June 1994
Limit. f(x) can be kept arbitrarily close to y by keeping x sufficiently close to $x_0$ (or, when $x_0 \longrightarrow \infty$, by keeping $x$ sufficiently large). The quantity on the right-hand side must be independent of the limit variable x. These notations should not be used for asymptotic forms.
$\lim\limits_{x \to x_0} f(x) = y$
$f(x) \to y \text{ as }x \to x_0$
$f(x) \mathop{\longrightarrow}\limits_{x\rightarrow x_0} y$
Asymptotic form. $f(x)/F(x) \to 1$ in the limit considered (usually $x \to \infty$). Contrast $f(x) = O(x^n)$ with $f(x) \sim Cx^n$ (C significant). The latter is an instance of the general notation $f(x) \sim \sum_{n}x^{-n} \Leftrightarrow \text{ for every }N, x^N[f(x) - \sum^{N}_n C_nx^{-n}] \to 0$
$f(x) \sim F(x)$
Limit superior, inferior. $y_1$ is the smallest upper limit, $y_2$ the largest lower limit, on $f \text{ as } x \to \infty$. Do not use merely for $y_2 \le f \le y_1$. The equivalent symbols $\overline{lim}, \underline{lim}$ may also be used.
lim sup $f(x) = y_1$
lim inf $f(x) = y_2$
Order. $f(x) \le\text{ const } × x^n$. Use = as the relation sign, not $\to$ or $\sim$. Does not mean “order of magnitude.” $f(x) = O(1)$ means merely that $|f| \le$ some nonzero constant. $f(x) = O(10^{−2})$ is meaningless: Use ~ instead. The O is set as an italic capital letter; it should not be looped or marked “script.”
$f(x) = O(x^n)$
“Little oh.” $f(x)/x^n \to 0$. The o is italic lowercase.
$f(x) = o(x^n)$
Order of magnitude. log$_{10} f \simeq n$.
$f(x) \sim 10^n$
Approximate equality. $\simeq$ is preferred, but $\cong$ and $\approx$ are acceptable. The symbols $\doteq$, $\doteqdot$, and $\bumpeq$ are ambiguous; if used, please provide definition in text.
$f(x) \simeq g(x)$