Question

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

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

Answer 1

Answer: (D)

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