Q108 / 180 · KCET 2014 · Mathematics
\[
\text { If } f(x)=\left\{\begin{array}{cc}
\frac{3 \sin \Pi x}{5 x} ; & x \neq 0 \\
2 K ; & x=0
\end{array}\right.
\]
is continuous at \( x=0 \), then the value of \( K \) is
is continuous at \( x=0 \), then the value of \( K \) is
- Correct answerA\( \frac{3 \pi}{10} \)
- B\( \frac{3 \pi}{5} \)
- C\( \frac{\Pi}{10} \)
- D\( \frac{3 \pi}{2} \)
Solution
Given that \( f(x)=\left\{\begin{array}{cl}\frac{3 \sin \pi x}{5 x} & x \neq 0 \\ 2 K & x=0\end{array}\right. \)
Now,
\[ \begin{array}{l} \lim _{x \rightarrow 0}\left(\frac{3 \sin \pi x}{5 x}\right)=2 K \\ \Rightarrow \frac{3 \pi}{5} \lim _{x \rightarrow 0} \frac{\sin \pi x}{x I I}=2 K \\ \Rightarrow \frac{3 \Pi}{5} \times 1=2 K \Rightarrow K=\frac{3 \Pi}{10} \end{array} \]
Now,
\[ \begin{array}{l} \lim _{x \rightarrow 0}\left(\frac{3 \sin \pi x}{5 x}\right)=2 K \\ \Rightarrow \frac{3 \pi}{5} \lim _{x \rightarrow 0} \frac{\sin \pi x}{x I I}=2 K \\ \Rightarrow \frac{3 \Pi}{5} \times 1=2 K \Rightarrow K=\frac{3 \Pi}{10} \end{array} \]
Hence, the correct answer is option: 'A'.