Let $f(x) = \left\{ \begin{array}{r} \frac{\sin{\pi x}}{5x},\ \ \ x
eq 0 \ k,\ \ \ \ \ \ x = 0 \end{array} ight.\$, if $f(x)$ is continuous at $x = 0$, then $k$ is equal to

Question

Let $f(x) = \left\{ \begin{array}{r} \frac{\sin{\pi x}}{5x},\ \ \ x 
eq 0 \ k,\ \ \ \ \ \ x = 0 \end{array} ight.\$, if $f(x)$ is continuous at $x = 0$, then $k$ is equal to

Dr. Abubacker · Accepted Answer

(a)
Since, $\lim_{x \rightarrow 0}\ \ f(x) = f(0)$
$\Rightarrow \ \lim_{x \rightarrow 0}\frac{\sin{\pi x}}{5x} = k\$
$\Rightarrow \ \ (1)\frac{\pi}{5} = k\ \ \  \Rightarrow \ \ \ k = \frac{\pi}{5}\ \ \$ $\left\lbrack \because\ \ \ \lim_{x \rightarrow 0}\ \frac{\sin x}{x} = 1 \right\rbrack$

Q. Let $f(x) = \left\{ \begin{array}{r} \frac{\sin{\pi x}}{5x},\ \ \ x \neq 0 \\ k,\ \ \ \ \ \ x = 0 \end{array} \right.\$, if $f(x)$ is continuous at $x = 0$, then $k$ is equal to

Explanation

Reviewed by Dr. Abubacker

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