Question

Calcule os limites, de maneira usual
lim
θ→0
sen8 θ/9θ

Answer 1

Limite trigonométrico fundamental

$\displaystyle\mathtt{\lim_{u \to 0}\dfrac{sen(u)}{u}=1}$

$\displaystyle\mathtt{\lim_{\theta \to 0}\dfrac{sen(8\theta)}{9\theta}}$

multiplicando e divindo por 8 temos

$\displaystyle\mathtt{\lim_{\theta \to 0}\dfrac{sen(8\theta)}{9\theta}. \dfrac{8}{8}}$

$\displaystyle\mathtt{\lim_{\theta \to 0}\dfrac{8}{9}\dfrac{sen(8\theta)}{8\theta}}$

Colocando a constante para fora do limite temos

$\displaystyle\mathtt{\dfrac{8}{9}\lim_{\theta \to 0}\dfrac{sen(8\theta)}{8\theta}}$

Vamos realizar uma substituição apropriada.

$\mathsf{u=8\theta~u\to~0~se~\theta\to\,0}$

Daí

$\displaystyle\mathtt{\dfrac{8}{9}\lim_{u \to 0}\dfrac{sen(u)}{u}=\dfrac{8}{9}.1=\dfrac{8}{9}}$