Question

Evaluate:

$\underset{x\to 0}{lim}{\left(\frac{\tan x}{x}\right)}^{1/x^3}$

• A. $\infty$
• B. 0
• C. 1
• D. $e^2$

Answer 1

The correct option is A $\infty$

limit is of the form $1^{\infty }$

Now, ${lim}_{x\to 0}{\left(\frac{\tan x}{x}\right)}^{1/x^3}$

$=e^{{lim}_{x\to 0}\left(\frac{\tan x-x}{x^4}\right)}$

$=e^{{lim}_{x\to 0}\left\{\frac{(x+\frac{x^3}{3}+\frac{2x^3}{15}+..\infty )-x}{x^4}\right\}}$

$=e^{{lim}_{x\to 0}\frac{1}{3x}}=\infty$
Ans: A