GATE CS 1995 Q7(B) — Calculus: Limits, Continuity, Differentiability

To evaluate the limit $\lim_{x \to 0} \frac{\sin x}{x}$ without using power series expansions, we can use two standard calculus approaches:

Method 1: L'Hôpital's Rule
Since substituting $x = 0$ yields the indeterminate form $\frac{0}{0}$, we can differentiate the numerator and the denominator with respect to $x$:

Method 2: Squeeze Theorem (Geometric Proof)
For a small positive angle $x$ in radians, comparing the areas of the inner triangle, the circular sector, and the outer triangle on a unit circle gives the inequality:

Dividing the inequality by $\sin x$ (since $\sin x > 0$ for $x \in (0, \pi/2)$):

Taking the limit as $x \to 0$, since $\lim_{x \to 0} \cos x = 1$, by the Squeeze Theorem, we get: