If limx→∞⁡1+ax-4x2^2x=e^3, then a is equal to

Correct answer: (B) 3/2. limx→∞1+ax-4x2^2x (1^(∞) form) =e^(limx→∞⁡1+ax-4x2-12x) =e^(limx→∞⁡2a-8x=e2a) ∴ e^2a=e^3 ⇒2a=3 ⇒ a=3/2

Q53 / 90 · JEE Main 2016 (09 Apr Online) · Mathematics

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

A
$2$
Correct answer
B
$\frac{3}{2}$
C
$\frac{1}{2}$
D
$\frac{2}{3}$

Solution

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

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

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

$∴$ $e^{2 a} = e^{3}$

$\Rightarrow 2 a = 3$

$\Rightarrow a = \frac{3}{2}$

Hence, the correct answer is option: 'B'.

If limx→∞⁡1+ax-4x22x=e3, then a is equal to

Solution

Relevant Questions

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