Question

Solve $\underset{x\to \pi /2}{lim}\frac{\sin x-{\left(\sin x\right)}^{\sin x}}{1-\sin x+\log \sin x}$

Answer 1

Substitute $\sin x=t$ and as $x\to \pi /2,t\to 1$
Also $\frac{d}{dx}\left(x^x\right)=x^x(1+\log x)$
$\therefore \underset{t\to 1}{lim}\frac{t-t^t}{1-t+\log t}\left(Form\frac{0}{0}\right)$
$=\underset{t\to 1}{lim}\frac{1-t^1(1+\log t)}{-1+\frac{1}{t}}\left(Form\frac{0}{0}\right)$
Apply L'Hospital's rule. $=\underset{t\to 1}{lim}\frac{-\{t^t.\frac{1}{t}+t^t{(1+\log t)}^2\}}{-1/t^2}=\frac{1+1}{1}=2$
Ans: 2