Ex 12.1

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Last updated at May 7, 2024 by Teachoo

Ex 12.1, 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/𝑠𝑖𝑛𝑏𝑥 Multiplying & dividing by ax = (𝒍𝒊𝒎)┬(𝒙→𝟎) 𝒔𝒊𝒏𝒂𝒙/𝒂𝒙 × (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛𝑏𝑥 = 1 × (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛𝑏𝑥 = (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛𝑏𝑥 Multiplying & dividing by bx = (𝑙𝑖𝑚)┬(𝑥→0) 𝑎𝑥/𝑠𝑖𝑛𝑏𝑥 × 𝑏𝑥/𝑏𝑥 Using lim┬(x→0) sin𝑥/𝑥 = 1 Replacing x by ax lim┬(x→0) sin𝑎𝑥/𝑎𝑥 = 1 = (𝑙𝑖𝑚)┬(𝑥→0) 𝑏𝑥/𝑠𝑖𝑛𝑏𝑥 × 𝑎𝑥/𝑏𝑥 = (𝑙𝑖𝑚)┬(𝑥→0) 𝑏𝑥/𝑠𝑖𝑛𝑏𝑥 × 𝑎/𝑏 = 𝑎/𝑏 "×" (𝑙𝑖𝑚)┬(𝑥→0) 𝑏𝑥/𝑠𝑖𝑛𝑏𝑥 = 𝑎/𝑏 ÷ (𝒍𝒊𝒎)┬(𝒙→𝟎) 𝒔𝒊𝒏𝒃𝒙/𝒃𝒙 = 𝑎/𝑏 ÷ 1 = 𝒂/𝒃 Using lim┬(x→0) sin𝑥/𝑥 = 1 Replacing x by bx lim┬(x→0) sin𝑏𝑥/𝑏𝑥 = 1