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Ex 13.1, 14 - Evaluate: lim x->0 sin ax/sin bx - Class 11 CBSE

Ex 13.1, 14 - Chapter 13 Class 11 Limits and Derivatives - Part 2

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Ex 13.1, 14 Evaluate the Given limit: lim┬(x→0) sin⁡〖ax 〗/(sin bx), a, b ≠ 0 (𝑙𝑖𝑚)┬(𝑥→0) 𝑠𝑖𝑛⁡〖𝑎𝑥 〗/(𝑠𝑖𝑛 𝑏𝑥) = (𝑙𝑖𝑚)┬(𝑥→0) sin ax × (𝑙𝑖𝑚)┬(𝑥→0) 1/𝑠𝑖𝑛⁡𝑏𝑥 Multiplying & dividing by ax = (𝒍𝒊𝒎)┬(𝒙→𝟎) 𝒔𝒊𝒏⁡𝒂𝒙/𝒂𝒙 × (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛⁡𝑏𝑥 = 1 × (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛⁡𝑏𝑥 = (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛⁡𝑏𝑥 Multiplying & dividing by bx = (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛⁡𝑏𝑥 × 𝑏𝑥/𝑏𝑥 Using lim┬(x→0) sin⁡𝑥/𝑥 = 1 Replacing x by ax lim┬(x→0) sin⁡𝑎𝑥/𝑎𝑥 = 1 = (𝑙𝑖𝑚)┬(𝑥→0) 𝑏𝑥/𝑠𝑖𝑛⁡𝑏𝑥 × 𝑎𝑥/𝑏𝑥 = (𝑙𝑖𝑚)┬(𝑥→0) 𝑏𝑥/𝑠𝑖𝑛⁡𝑏𝑥 × 𝑎/𝑏 = 𝑎/𝑏 "×" (𝑙𝑖𝑚)┬(𝑥→0) 𝑏𝑥/𝑠𝑖𝑛⁡𝑏𝑥 = 𝑎/𝑏 ÷ (𝒍𝒊𝒎)┬(𝒙→𝟎) 𝒔𝒊𝒏⁡𝒃𝒙/𝒃𝒙 = 𝑎/𝑏 ÷ 1 = 𝒂/𝒃 Using lim┬(x→0) sin⁡𝑥/𝑥 = 1 Replacing x by bx lim┬(x→0) sin⁡𝑏𝑥/𝑏𝑥 = 1

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.