Question

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

• A. $0$
• B. $1$
• C. $e$
• D. $Doesnotexist$

Answer 1

The correct option is D $Doesnotexist$

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

$=e^{{lim}_{x\to 0}(\frac{\tan x}{x}-1)\left(\frac{1}{x^3}\right)}$

$=e^{{lim}_{x\to 0}\left(\frac{\tan x-x}{x}\right)\left(\frac{1}{x^3}\right)}$

$=e^{{lim}_{x\to 0}\left(\frac{1}{x^4}\right)\left(x+\frac{x^3}{3}+\frac{2x^5}{15}+....-x\right)}$

$=e^{{lim}_{x\to 0}\left(\frac{1}{x^4}\right)\left(\frac{x^3}{3}+\frac{2x^5}{15}+....\right)}$

$=e^{{lim}_{x\to 0}\left(\frac{1}{3x}+\frac{2x}{15}+....\right)}$

$=e^{\infty }$

Does not exist

Option D is correct.