http://www.bilibili.com/mobile/video/av2261356.html

[例子] $p (x) = a + bx+cx^{2}+dx^{3}$

$p(x) - tan x \sim x^{3}, \text { when } x \to 0$

Find a, b, c, d ?

[Solution] :

1. Don’t use l’Hôpital Rule for  $\displaystyle \lim \frac {f}{g}$

2. Apply Taylor expansion :

$\tan x = x + \frac {1}{3}x^{3} + o (x^{3})$
$p (x) - \tan x = a + (b -1)x + (c - \frac {1}{3})x^{3} + o (x^{3})$

$p(x) - tan x \sim x^{3}, \text { when } x \to 0$

$\iff \boxed {a=0, b=1, c=\frac {4}{3}}$