Question

qual o valor do limite da questão abaixo ???

Answer 1

Explicação passo-a-passo:

$\sf \lim_{x~\to~+\infty}~\Big(\dfrac{3}{4}\Big)^x$

$\sf =\lim_{x~\to~+\infty}~\dfrac{3^x}{4^x}$

$\sf =\lim_{x~\to~+\infty}~\dfrac{\frac{3^x}{3^x}}{\frac{4^x}{3^x}}$

$\sf =\lim_{x~\to~+\infty}~\dfrac{1}{\frac{4^x}{3^x}}$

$\sf =\lim_{x~\to~+\infty}~\dfrac{1}{\Big(\frac{4}{3}\Big)^x}$

Note que $\sf \dfrac{4}{3} > 1$, logo:

$\sf \lim_{x~\to~+\infty}~\Big(\dfrac{4}{3}\Big)^x=+\infty$

Assim:

$\sf \lim_{x~\to~+\infty}~\dfrac{1}{\Big(\frac{4}{3}\Big)^x}=0$

Portanto:

$\sf \red{\lim_{x~\to~+\infty}~\Big(\dfrac{3}{4}\Big)^x=0}$