Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{1-{\sin }^{3}x}{{\cos }^{2}x}$

Answer 1

$\underset{x\to \frac{\pi }{2}}{lim}\frac{1-{\sin }^{3}x}{{\cos }^{2}x}\left(\frac{0}{0}form\right)$

Using L'Hospital Rule

$=\underset{x\to \frac{\pi }{2}}{lim}\frac{-3{\sin }^{2}x\mathrm{cosx}}{-2\cos x\sin x}$

$\underset{x\to \frac{\pi }{2}}{=lim}\frac{3\sin x}{2}$

$=\frac{3}{2}$