Ex 12.1, 20 - Chapter 12 Class 11 Limits and Derivatives

Transcript

Ex 12.1, 20 Evaluate the Given limit: lim┬(x→0) (𝑠𝑖𝑛⁡𝑎𝑥 + 𝑏𝑥)/(𝑎𝑥 + 𝑠𝑖𝑛⁡𝑏𝑥 ) a , b, a + b ≠ 0 lim┬(x→0) (𝑠𝑖𝑛⁡𝑎𝑥 + 𝑏𝑥)/(𝑎𝑥 +〖 𝑠𝑖𝑛〗⁡𝑏𝑥 ) = lim┬(x→0) 𝑥(𝑠𝑖𝑛⁡𝑎𝑥/𝑥 + 𝑏)/𝑥(𝑎 + 𝑠𝑖𝑛⁡𝑏𝑥/𝑥) = lim┬(x→0) ((𝑠𝑖𝑛⁡𝑎𝑥/𝑥 ) + 𝑏)/(𝑎 + ( 𝑠𝑖𝑛⁡𝑏𝑥/𝑥) ) Multiply & Divide by 𝑠𝑖𝑛⁡𝑎𝑥/𝑥 by ax & Multiply & Divide 𝑠𝑖𝑛⁡〖𝑥 〗/𝑏 by bx = lim┬(x→0) ((𝑠𝑖𝑛⁡𝑎𝑥/𝑥 . 𝑎𝑥/𝑎𝑥 ) + 𝑏)/(𝑎 + ( 𝑠𝑖𝑛⁡𝑏𝑥/𝑥 . 𝑏𝑥/𝑏𝑥) ) = lim┬(x→0) ((𝑠𝑖𝑛⁡𝑎𝑥/𝑎𝑥 . 𝑎𝑥/𝑥 ) + 𝑏)/(𝑎 + ( 𝑠𝑖𝑛⁡𝑏𝑥/𝑏𝑥 . 𝑏𝑥/𝑥) ) = lim┬(x→0) ((𝒔𝒊𝒏⁡𝒂𝒙/𝒂𝒙). 𝑎 + 𝑏)/(𝑎 + ( 𝒔𝒊𝒏⁡𝒃𝒙/𝒃𝒙) 𝑏) Using lim┬(x→0) (sin⁡x )/x = 1 Replacing x by ax. lim┬(x→0) sin⁡𝑎𝑥/ax = 1 Replacing x by bx lim┬(x→0) (sin⁡bx )/bx = 1 = ((𝟏) 𝑎 + 𝑏)/(𝑎 +(𝟏)𝑏) = (𝑎 + 𝑏)/(𝑎 + 𝑏) = 1