Why lim (x→0) sin x / x = 1: comparing areas in a unit circlelim (x→0) sin x / x = 1 क्यों: इकाई वृत्त में क्षेत्रफलों की तुलनाWhy lim (x→0) sin x / x = 1: comparing areas in a unit circle

Section 12.4 · Class 11 Maths Ch 12 · interactive diagram

Parts and what they mean

• Centre O (केंद्र O) — Unit circle: radius OA = 1.
• A, B, C (A, B, C) — B on the circle at angle x; C where OB extended meets the tangent at A.
• Angle x (radians) (कोण x (रेडियन)) — Take 0 < x < π/2, measured in radians.
• sin x (sin x) — Height of B. Area of triangle OAB = ½ sin x.
• Sector OAB (त्रिज्यखंड OAB) — Arc length = x; sector area = ½ x.
• tan x (tan x) — AC = tan x. Area of triangle OAC = ½ tan x.
• Sandwich (squeeze) (सैंडविच प्रमेय) — ½ sin x < ½ x < ½ tan x ⇒ cos x < sin x / x < 1. As x → 0, cos x → 1, so sin x / x → 1.

Original diagram drawn by Simple Learning Hub for NCERT Class 11 Maths Ch 12. Report a mistake