The function $f(x) =\left\{\begin{matrix}\frac{\sin 2x}{x}+\cos x,&\text{if x≠0}\\K,&\text{if x=0}\end{matrix}\right.$ is continuous at $x = 0$, then the value of $K$ is:

3

The correct answer is Option (4) → 3

$f(x)= \begin{cases} \frac{\sin 2x}{x}+\cos x, & x\neq 0 \\ K, & x=0 \end{cases}$

For continuity at $x=0$,

$\lim_{x\to 0} f(x) = f(0) = K$

$\lim_{x\to 0}\left(\frac{\sin 2x}{x}+\cos x\right) = \lim_{x\to 0}\frac{\sin 2x}{x} + \lim_{x\to 0}\cos x$

$= \lim_{x\to 0}\frac{\sin 2x}{2x}\cdot 2 + 1$

$= 1\cdot 2 + 1 = 3$

$therefore K = 3$