GATE CS 2008 Q1 — Calculus: Limits, Continuity, Differentiability

To evaluate the limit:

$$\lim_{x \to \infty} \frac{x - \sin x}{x + \cos x}$$

We divide both the numerator and the denominator by $x$ (since $x \neq 0$ as $x \to \infty$):

$$\lim_{x \to \infty} \frac{1 - \frac{\sin x}{x}}{1 + \frac{\cos x}{x}}$$

Now, let us analyze the terms $\frac{\sin x}{x}$ and $\frac{\cos x}{x}$ as $x \to \infty$:

• Since $-1 \le \sin x \le 1$, the term $\sin x$ is bounded. Therefore, by the Squeeze Theorem: $$\lim_{x \to \infty} \frac{\sin x}{x} = 0$$
• Similarly, since $-1 \le \cos x \le 1$, the term $\cos x$ is also bounded: $$\lim_{x \to \infty} \frac{\cos x}{x} = 0$$

Substituting these limits back into the expression:

$$\lim_{x \to \infty} \frac{1 - \frac{\sin x}{x}}{1 + \frac{\cos x}{x}} = \frac{1 - 0}{1 + 0} = 1$$

Thus, the correct option is A.