Step 1 (Formula / Governing Law & Given Data): [1/4 Mark]

For a function $f(x)$ to be continuous at $x = a$, the limit of the function as $x$ approaches $a$ must be equal to the value of the function at $a$. Mathematically:

$$\lim_{x \to a} f(x) = f(a)$$

Given $a = 0$, we must satisfy: $\lim_{x \to 0} f(x) = f(0) = k$.

Step 2 (Step-by-Step Mathematical Working): [1/2 Mark]

Substitute the given function into the limit definition:

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

Apply the limit properties (sum rule):

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

Using standard limits ($\lim_{x \to 0} \frac{\sin x}{x} = 1$) and direct substitution for $\cos x$:

$$1 + \cos(0) = k$$ $$1 + 1 = k$$

Step 3 (Final Concluded Answer): [1/4 Mark]

Therefore, the value of $k$ is:

$$k = 2$$

The correct option is (D) 2.

💡 EL10 Concept Insight (Easy to Understand):

Think of continuity like drawing a line without lifting your pencil; for the line to be unbroken at zero, the "path" coming from both sides must land exactly on the point $k$. Since the path approaches 2, $k$ must be 2 to "fill the hole" and keep the line connected.