If _ x → ∞ (1 + (a)/(x) - 4 x ^ 2 ) ^ 2x = e ^ 3 , then a is equal...

\lim_{x \to \infty}{\left(1 + \dfrac{a}{x} - \dfrac{4}{{x}^{2}}\right)}^{2x} $( {1}^{\infty}$form$)$

$= {e}^{\left[\lim_{x \to \infty} \left(1 + \dfrac{a}{x} - \dfrac{4}{{x}^{2}} - 1\right)2x\right]}$

$= {e}^{\lim_{x \to \infty} \left(2a - \dfrac{8}{x}\right) = {e}^{2a}}$

$∴$ ${e}^{2a} = {e}^{3}$

$⇒ 2a = 3$

$⇒ a = \dfrac{3}{2}$