The value of _ x → ∞ e ^ x + 1 ( x ^ 3 e ^ - x + 1) si n ^ 3 (2x)...

Given, $\lim_{x \to \infty}\dfrac{e ⋅ {e}^{x}\log \left(\dfrac{{x}^{3}}{{e}^{x}} + 1\right)}{{\left(\sin 2x\right)}^{3}}$

$= \lim_{x \to \infty}\dfrac{elog\left(\dfrac{{x}^{3}}{{e}^{x}} + 1\right)}{\left(\dfrac{{x}^{3}}{{e}^{x}}\right)} \times \dfrac{{\left(2x\right)}^{3}}{{\left(\sin 2x\right)}^{3}} \times \dfrac{1}{8}$

$= e \times 1 \times 1 \times \dfrac{1}{8} = \dfrac{e}{8} \left(∵ \dfrac{{x}^{3}}{{e}^{x}} \to 0ifx \to \infty\right)$