Question

For what value of k is the function

\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]  continuous at x = 0?

 

Answer 1

Given: 

\[f\left( x \right) = \binom{\frac{\sin2x}{x}, x \neq 0}{k, x = 0}\]

If f(x) is continuous at x = 0, then

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

\[\Rightarrow \lim_{x \to 0} \frac{\sin2x}{x} = k\]

\[\Rightarrow \lim_{x \to 0} \frac{2\sin2x}{2x} = k\]

\[\Rightarrow 2 \lim_{x \to 0} \frac{\sin2x}{2x} = k\]

\[\Rightarrow 2 \times 1 = k\]

\[\Rightarrow k = 2\]