Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{1-{\left(\sin x\right)}^{1/3}}{1-{\left(\sin x\right)}^{2/3}}$

Answer 1

$L={lim}_{x\to \frac{\pi }{2}}\frac{1-{\left(\sin x\right)}^{\frac{1}{3}}}{1-{\left(\sin x\right)}^{\frac{2}{3}}}$

Applying L' Hospitals rule we have,

$={lim}_{x\to \frac{\pi }{2}}\frac{\frac{-1}{3}{\left(\sin x\right)}^{\frac{-2}{3}}.\cos x}{\frac{-2}{3}{\left(\sin x\right)}^{\frac{-1}{3}}.\cos x}={lim}_{x\to \frac{\pi }{2}}\frac{1}{2}\times \frac{1}{{\left(\sin x\right)}^{\frac{1}{3}}}L={lim}_{x\to \frac{\pi }{2}}\frac{1}{2}\times \frac{1}{{\left(\sin x\right)}^{\frac{1}{3}}}=\frac{1}{2}\times \frac{1}{{\left(\sin \frac{\pi }{2}\right)}^{\frac{1}{3}}}=\frac{1}{2}$