Question

The value of ${lim}_{x\to 0}\left(\right(sinx{)}^{\frac{1}{x}}+\left(1/x{)}^{sinx}\right)$ where $x>0$ is

• A. \N
• B. -1
• C. 1
• D. 2

Answer 1

The correct option is C 1

${lim}_{x\to 0}(sinx{)}^{\frac{1}{x}}+{\left(\frac{1}{x}\right)}^{sinx}={lim}_{x\to 0}(sinx{)}^{\frac{1}{x}}+{lim}_{x\to 0}{\left(\frac{1}{x}\right)}^{sinx}=0+e^{{lim}_{x\to 0}sinxln\frac{1}{x}}=e^{{lim}_{x\to 0}\frac{-ln\left(x\right)}{cosecx}}=e^{{lim}_{x\to 0}\frac{-\frac{1}{x}}{-cosecxcotx}}=e^{{lim}_{x\to 0}\frac{sinx}{x}.tanx}=e^0=1$