Question

For $x>0,\underset{x\to 0}{lim}\left(\right(\sin x{)}^{1/x}+\left(1/x{)}^{\sin x}\right)$ is

• A. $0$
• B. $-1$
• C. $1$
• D. $2$

Answer 1

Answer: (C)

$1$

${lim}_{x\to 0}({\left(\sin x\right)}^{1/x}+{\left(\frac{1}{x}\right)}^{smx})$

$=0+e^{\underset{x\to 0}{lim}sinx\ln \left(\frac{1}{x}\right)}=e^{-\underset{x\to 0}{lim}\frac{\ln x}{\csc x}}$

(Using $L$' Hospital's rule).
$=e^{-\underset{x\to 0}{lim}\frac{1/x}{-\csc x.\cot x}}=e^{-\underset{x\to 0}{lim}\frac{{\sin }^{2}x}{x.\cos x}}$

$=e^{-\underset{x\to 0}{lim}\frac{x.{\sin }^{2}x}{x^2.\cos x}}=e^0=1$