Question

The $\displaystyle\lim_{y \to a} \left\{ \left(\sin \frac{y-a}{2}\right) . \left(\tan \frac{\pi y}{2a}\right)\right\} $ is

• A. $\frac{a}{2 \pi}$
• B. $\frac{2 a}{ \pi}$
• C. $\frac{a}{\pi}$
• D. $ - \frac{a}{\pi}$

Answer 1

The Correct Option is D

Let $L =\displaystyle\lim _{y \rightarrow a}\left\{\left(\sin \frac{y-a}{2}\right)\left(\tan \frac{\pi y}{2 a}\right)\right\} $
$=\displaystyle\lim _{y \rightarrow a} \frac{\sin \frac{y-a}{2}}{\cot \frac{\pi y}{2 a}} $
$[\frac{0}{0} $ form ]
Using by L'Hospital rule, we get
$=\displaystyle\lim _{y \rightarrow a} \frac{\frac{1}{2} \cos \frac{y-a}{2}}{-\frac{\pi}{2 a}
\text{cosec}^{2} \frac{\pi y}{2 a}} $
$=\frac{\frac{1}{2} \times 1}{-\frac{\pi}{2 a} \cdot 1}=\frac{-a}{\pi}$