Limite Trigonométrico

por **jmoura** » Seg Mar 26, 2012 03:34

Como resolvo esse limite?

$\lim_{x->0} \frac{sen(x).sen(3x).sen(5x)}{tan(2x).tan(4x).tan(6x)}$

por **LuizAquino** » Seg Mar 26, 2012 17:24

jmoura escreveu:Como resolvo esse limite?

$\lim_{x->0} \frac{sen(x).sen(3x).sen(5x)}{tan(2x).tan(4x).tan(6x)}$

Note que:

$\lim_{x\to 0} \dfrac{\,\textrm{sen}\,x \,\textrm{sen}\,3x \,\textrm{sen}\, 5x}{\,\textrm{tg}\,2x \,\textrm{tg}\,4x \,\textrm{tg}\, 6x} = \lim_{x\to 0} \frac{\,\textrm{sen}\,x \,\textrm{sen}\,3x \,\textrm{sen}\, 5x}{\frac{\,\textrm{sen}\,2x}{\cos 2x} \frac{\,\textrm{sen}\,4x}{\cos 4x} \frac{\,\textrm{sen}\,6x}{\cos 6x}}$

$= \lim_{x\to 0} \dfrac{\cos 2x \cos 4x \cos 6x\,\textrm{sen}\,x \,\textrm{sen}\,3x \,\textrm{sen}\, 5x}{\textrm{sen}\,2x \,\textrm{sen}\,4x \,\textrm{sen}\,6x}$

$= \lim_{x\to 0} \dfrac{(\cos 2x \cos 4x \cos 6x) (x)(3x)(5x)\dfrac{\,\textrm{sen}\,x \,\textrm{sen}\,3x \,\textrm{sen}\, 5x}{(x)(3x)(5x)}}{(2x)(4x)(6x)\dfrac{\textrm{sen}\,2x \,\textrm{sen}\,4x \,\textrm{sen}\,6x}{(2x)(4x)(6x)}}$

Agora tente terminar o exercício.