Question

$\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x^2}$ is equal to

• A. $e^{1/3}$
• B. $1$
• C. $e$
• D. $0$

Answer 1

The correct option is A $e^{1/3}$

$L=\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x^2}\left[1^{\infty }\text{form}\right]$
$=\underset{x\to 0}{lim}{(1+\frac{\tan x}{x}-1)}^{1/x^2}$
$=\exp \left(\underset{x\to 0}{lim}(\frac{\tan x}{x}-1)\frac{1}{x^2}\right)$
$=\exp \left(\underset{x\to 0}{lim}\frac{\tan x-x}{x^3}\right)$
$=\exp \left(\underset{x\to 0}{lim}\frac{x+\frac{1}{3}{x}^3+\frac{2}{15}{x}^5+\cdots -x}{x^3}\right)$
$=\exp \left(\underset{x\to 0}{lim}\frac{\frac{1}{3}{x}^3+\frac{2}{15}{x}^5+\cdots }{x^3}\right)$
$=e^{1/3}$