Question

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

• A. $\underset{x\to \frac{{\pi }^{-}}{}}{\lim }f(x)=-\infty$
• B. $\underset{x\to \frac{{\pi }^{+}}{2}}{\lim }f(x)=\infty$
• C. $\underset{x\to \frac{\pi }{2}}{\lim }f(x)=\infty$
• D. none of these

Answer 1

Answer: (D)

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