Question

Let $f(x)=\underset{h\to 0}{\lim }\frac{(\sin (x+h){)}^{\ln (x+h)}-(\sin x{)}^{\ln x}}{h}$, then $f\left(\frac{\pi }{2}\right)$ is

• A. equal to 0
• B. equal to 1
• C. $\ln \frac{\pi }{2}$
• D. non-existent

Answer 1

Answer: (A)

Let $g(x)=(\sin x{)}^{\ln x}=e^{\ln x\cdot \ln (\sin x)}$
$f(x)=g^{\prime }(x)=(\sin x{)}^{\ln x}\left[\cot x(\ln x)+\frac{\ln (\sin x)}{x}\right]$
Hence, $f\left(\frac{\pi }{2}\right)=g^{\prime }\left(\frac{\pi }{2}\right)=1(0+0)=0$