Asymptotics: Introduction II

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We say that $f$ is a ‘Big O’ of $g$ as $x \to x_0$ and write

$\displaystyle f(x) = \mathcal{O}(g(x)) \text{ if } \frac{f(x)}{g(x)} \text{ is bounded as } x \to x_0$

Here are a few simple examples:

$\sin(x) = x + \mathcal{O}(x^3)$ as $x \to 0$ since $\displaystyle\lim_{x \to 0}\frac{\sin(x)-x}{x^3} = -\frac{1}{6}$
$\cos(x) = 1 + \mathcal{O}(x^2)$ as $x \to 0$ since $\displaystyle\lim_{x \to 0}\frac{\cos(x)-1}{x^2} = -\frac{1}{2}$
$e^x = 1 + x + \mathcal{O}(x^2)$ as $x \to 0$ since $\displaystyle\lim_{x \to 0}\frac{e^x-1-x}{x^2} = \frac{1}{2}$

We say that $f$ is a ‘Little O’ of $g$ as $x \to x_0$ and write

$\displaystyle f(x) = o(g(x)) \text{ if } \lim_{x \to x_0} \frac{f(x)}{g(x)} = 0$

Alternatively, we can write

$\displaystyle f(x) \ll g(x)$

Here are a few simple examples:

Big O	Little o
$\displaystyle f(x)=\mathcal{O}(g(x))$	$\displaystyle f(x)=o(g(x))$
The quotient $\displaystyle\frac{f(x)}{g(x)}$ is bounded	The quotient $\displaystyle\frac{f(x)}{g(x)}$ tends to zero
f is at most of the same order as g	f is negligible compared to g

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