The function \(f(x) = \begin{cases}\displaystyle\frac{\sin 2x}{x}+\cos x, & \text{if } x \ne 0\\K, & \text{if } x = 0\end{cases}\) is continuous at \(x = 0\), then the value of \(K\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3
For continuity at x = 0, the limit of f(x) as x → 0 must equal f(0) = K.
\( \displaystyle \lim_{x \to 0}\left(\frac{\sin 2x}{x} + \cos x\right) = \lim_{x \to 0} 2\cdot\frac{\sin 2x}{2x} + \lim_{x \to 0}\cos x \)
Using the standard limit \(\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1\):
\( = 2(1) + 1 = 3 \)
So K = 3.