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Question
For what value of k is the function
\[f\left( x \right) = \begin{cases}\frac{\sin 5x}{3x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0?\]
Solution
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
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[\lim_{x \to 0} \frac{\sin5x}{3x} = k\]
\[\lim_{x \to 0} \frac{5 \sin5x}{3 \times 5x} = k\]
\[\frac{5}{3} \lim_{x \to 0} \frac{\sin5x}{5x} = k\]
\[\frac{5}{3} \times 1 = k\]
\[k = \frac{5}{3}\]
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