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Question

Mathematics Question on Limits

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

A

a2π\frac{a}{2 \pi}

B

2aπ\frac{2 a}{ \pi}

C

aπ\frac{a}{\pi}

D

−aπ - \frac{a}{\pi}

Answer

−aπ - \frac{a}{\pi}

Explanation

Solution

Let L =\displaystyle\lim _{y \rightarrow a}\left\\{\left(\sin \frac{y-a}{2}\right)\left(\tan \frac{\pi y}{2 a}\right)\right\\}
=lim⁡y→asin⁡y−a2cot⁡πy2a=\displaystyle\lim _{y \rightarrow a} \frac{\sin \frac{y-a}{2}}{\cot \frac{\pi y}{2 a}}
[00[\frac{0}{0} form ]
Using by L'Hospital rule, we get
=lim⁡y→a12cos⁡y−a2−π2acosec2πy2a=\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}}
=12×1−π2a⋅1=−aπ=\frac{\frac{1}{2} \times 1}{-\frac{\pi}{2 a} \cdot 1}=\frac{-a}{\pi}