Question

If $f\left(x\right)=\frac{\cos x}{(1-\sin x{)}^{1/3}}$, then

• A. $\underset{x\to \frac{\pi }{2}}{lim}f\left(x\right)=-\infty$
• B. $\underset{x\to \frac{\pi }{2}}{lim}f\left(x\right)=1$
• C. $\underset{x\to \frac{\pi }{2}}{lim}f\left(x\right)=\infty$
• D. None of these

Answer 1

Answer: (D)

None of these

Given, $f\left(x\right)=\frac{\cos x}{(1-\sin x{)}^{1/3}}$
$\underset{x\to \frac{\pi }{2}}{lim}\frac{\cos x}{(1-\sin x{)}^{1/3}}$
${lim}_{h\to 0}\frac{\cos (\frac{\pi }{2}+h)}{(1-\sin (\frac{\pi }{2}+h){)}^{1/3}}$
$=\underset{h\to 0}{lim}\frac{-\sin h}{(1-\cos h{)}^{1/3}}$
$=-\underset{h\to 0}{lim}\frac{2\sin \frac{h}{2}\cos \frac{h}{2}}{{\left(2{\sin }^2\frac{h}{2}\right)}^{1/3}}$
$=-\underset{h\to 0}{lim}{2}^{2/3}\cos \frac{h}{2}{\left(\sin \frac{h}{2}\right)}^{1/3}=0$