Question

For what value of k is the function 

\[f\left( x \right) = \begin{cases}\frac{\sin 5x}{3x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0?\]

Answer 1

Given:

\[f\left( x \right) = \binom{\frac{\sin5x}{3x}, if x \neq 0}{k, if x = 0}\]

If

\[f\left( x \right)\] is continuous at x = 0, then

\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]

\[\lim_{x \to 0} \frac{\sin5x}{3x} = k\]

\[\lim_{x \to 0} \frac{5 \sin5x}{3 \times 5x} = k\]

\[\frac{5}{3} \lim_{x \to 0} \frac{\sin5x}{5x} = k\]

\[\frac{5}{3} \times 1 = k\]

\[k = \frac{5}{3}\]