Question

If $\displaystyle\lim_{x \to \infty} \left( 1 + \frac{a}{x} - \frac{4}{x^2} \right)^{2x} = e^3$ , then $'a'$ is equal to :

• A. $2$
• B. $\frac{3}{2}$
• C. $\frac{2}{3}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

$Lim\left(1+\frac{a}{2}-\frac{4}{x^{2}}\right)^{2x}\left(1\infty form\right)$
$=e^{\left[\text{}\lim_{x \to \infty}\left(1+\frac{a}{x} \frac{4}{x^{2}}-1\right)2x \right] }$
$=e \displaystyle \lim_{x \to \infty}$$\left(2a-\frac{8}{x}\right)=e^{2a}$
$\therefore 2a=3 \Rightarrow a=3/2$