Question

If $\underset{x\to \frac{\pi }{2}}{lim}\frac{1-{\sin }^{3}x}{{\cos }^{2}x}$ is $\left(\frac{m}{n}\right)$.Then, find $(m+n)$

Answer 1

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

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

$=\underset{x\to \frac{\pi }{2}}{Lt}\frac{(1-\sin x)(1+\sin x+{\sin }^{2}x)}{(1-\sin x)(1+\sin x)}$...............using formula $m^3+n^3=(m+n)(m^2+mn+n^2)$

$=\underset{x\to \frac{\pi }{2}}{Lt}\frac{1+\sin x+{\sin }^{2}x}{1+\sin x}$

$=\frac{1+1+1}{1+1}$

$=\frac{3}{2}$

$So,m=3andn=2$

$\therefore m+n=5$