$\lim_{x \to \infty} \frac{x + \sin x}{x + \cos x}$
Step-by-Step Solution
Key Concept: Divide numerator and denominator by $x$: $\frac{1 + (\sin x)/x}{1 + (\cos x)/x}$. As $x \to \infty$, both $(\sin x)/x$ and $(\cos x)/x$ squeeze to 0 (bounded numerator, $x \to \infty$). Limit $= 1/1 = 1$.
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Correct Answer: (2)