Question

Evaluate: $\underset{x\to 0}{lim}\frac{\tan x-\sin x}{x^3}$

Answer 1

$li{m}_{x\to 0}\frac{tanx-sinx}{x^3}$

$=li{m}_{x\to 0}\frac{se{c}^{2}x-cosx}{3x^2}$

$=li{m}_{x\to 0}\frac{2se{c}^{2}xtanx+sinx}{6x}$

$=li{m}_{x\to 0}\frac{2se{c}^{4}x+4se{c}^{2}xta{n}^{2}x+cosx}{6}$

$=\frac{3}{6}=\frac{1}{2}$