A function $f(x)$ is continuous at $x=a$ if the limit of the function as $x$ approaches $a$ is equal to the value of the function at $a$. Here, we require $\lim_{x \to 0} f(x) = f(0)$.
We need to calculate the limit of the function as $x$ approaches $0$ for the case $x \neq 0$: $$ \lim_{x \to 0} \left( \frac{\sin x}{x} + \cos x \right) $$
Using the property of limits, we can split the expression: $$ \lim_{x \to 0} \frac{\sin x}{x} + \lim_{x \to 0} \cos x $$ We know that $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \cos x = \cos(0) = 1$.
Adding the results: $$ 1 + 1 = 2 $$ Since the function is continuous at $x=0$, $f(0) = k$. Therefore, $k = 2$.
Final Answer: 2
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