The value of k, for which the function \(f(x) = \begin{cases} \displaystyle\frac{\sin kx}{x} + 3\cos x, & x \ne 0 \\ 7, & x = 0 \end{cases}\) is continuous at x = 0, is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 4
f is continuous at x = 0 when \(\lim_{x \to 0} f(x) = f(0) = 7\).
Using the standard limit \(\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1\):
\( \displaystyle \lim_{x \to 0}\frac{\sin kx}{x} = \lim_{x \to 0} k \cdot \frac{\sin kx}{kx} = k \)
\( \displaystyle \lim_{x \to 0} 3\cos x = 3 \)
So
\(k + 3 = 7\)
\(k = 4\)