* Factorials are exceeded in growth rate only by _double exponential functions_ (e.g. $a_n = n^n \space \text{or} \space b_n = e^{x^c}$).
$$ \text{For instance, if} \space \{a_n\} = \frac{n!}{n^n} \space \text{, then} \space \lim_{n\to\infty} \{a_n\} = 0 $$As $n\to\infty \text{,}$ $\{a_n\} \to 0$ because $n^n$ grows faster than $n!$.
More details about growth rates and other data about factorials can be found on Wikipedia.
The highest power term dominates for polynomials/poly-types near $\infty$ and $-\infty$
If the highest-degree term of $f(x)$ is $ax^n$, then $f(x) \sim ax^n$ as $x \to \infty$ or $x \to -\infty$.
The lowest power dominates for polynomials/poly-types near zero (0).
If the lowest-degree term of $f(x)$ is $bx^m$, then $f(x) \sim bx^m$ as $x \to 0$.
Remember (important): exponentials grow faster than polynomials.
This means that, for example:
$$\lim_\limits{x\to \infty}\frac{x^n}{e^x} = 0$$
You can also write that as $e^{-x}x^n$, and divide by $x^n$, to get:
$e^{-x} \le \frac{C}{x^n}$ for all $x > 0$
Logs grow more slowly than any other function of $x$, near infinity. In fact, for any positive number $a > 0$, no matter how small, you have:
$$\lim_\limits{x\to \infty} \frac{\ln(x)}{x^a} = 0$$
Similarly,
$$\ln(x) \le Cx^a \text{ for all } x > 1$$
Always remember that:
$$\lim_\limits{x\to 0^+}x^a\ln(x) = 0$$
* "Poly-type" functions: functions involving terms with rational exponents, like the example below, are not technically polynomial functions. But their behavior at the extremes is similar to polynomials, so I've grouped them together in terms of behavior.
$$f(x) = 2x^5+4x^2+17x^{2/3}-12 \quad \quad g(x) = \sqrt{x^9-27x^3+81} \quad \quad h(x) = x^8 - \sqrt{x^2-\sqrt[4]{x^2+3}}$$
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