Compute without using power series expansion: lim_{x->0} sin(x) / x is __________.

GATE 1995 · Engineering Mathematics · Limits · medium

Answer: 1

1. Area inequality from unit circle: Inner triangle area <= sector area <= outer triangle area for 0 < x < pi/2
2. Convert to inequality for sin(x)/x: Equivalently: cos(x) <= sin(x)/x <= 1 for 0 < x < pi/2
3. Apply squeeze theorem: Since cos(x) <= sin(x)/x <= 1 and both bounds -> 1, by squeeze theorem lim_{x->0} sin(x)/x = 1