Let $k \in \mathbb{R}$. If $\lim_{x \to 0^+} (\sin(\sin kx) + \cos x + x)^{\frac{2}{x}} = e^{6}$, then the value of $k$ is

$\text{Given: } \lim_{x \to 0^+}\big(\sin(\sin kx) + \cos x + x\big)^{\frac{2}{x}} = e^6$

$\text{To find: } k \text{ and the correct option}$

$\text{Base} \to \sin 0 + \cos 0 + 0 = 1 \Rightarrow 1^{\infty} \text{ form}$

$\lim (1+f)^{g} = e^{\lim fg} \quad (\text{as } f \to 0)$

$f = \sin(\sin kx) + \cos x + x - 1, \qquad g = \dfrac{2}{x}$

$\sin(\sin kx) \approx \sin(kx) \approx kx, \quad \cos x \approx 1 - \dfrac{x^2}{2}$

$f \approx kx + x - \dfrac{x^2}{2} = (k+1)x + O(x^2)$

$fg \approx (k+1)x \cdot \dfrac{2}{x} = 2(k+1)$

$2(k+1) = 6 \Rightarrow k + 1 = 3 \Rightarrow k = 2$

$k = 2 \Rightarrow \text{option (2)}$

$\boxed{k = 2}$