Ex 12.1, 17 - Chapter 12 Class 11 Limits and Derivatives

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Ex 12.1, 17 Evaluate the Given limit: lim┬(x→0) cos⁡〖2x − 1〗/cos⁡〖x − 1〗 lim┬(x→0) ( 𝐜𝐨𝐬⁡〖𝟐𝐱 〗− 1)/cos⁡〖x − 1〗 = lim┬(x→0) ((𝟏 − 𝟐 〖𝐬𝐢𝐧^𝟐〗⁡𝒙) − 1)/cos⁡〖𝑥 − 1〗 = lim┬(x→0) (1 − 2 〖𝐬𝐢𝐧^𝟐〗⁡𝒙 − 1 )/cos⁡〖x − 1〗 = lim┬(x→0) (−2 𝐬𝐢𝐧𝟐 𝐱 )/cos⁡〖x − 1〗 = lim┬(x→0) (−2(𝟏 − 𝐜𝐨𝐬^𝟐 𝒙))/cos⁡〖x − 1〗 = lim┬(x→0) (−2(1 − 𝐜𝐨𝐬^𝟐 𝒙) )/(−1(1−〖 cos〗⁡〖𝑥)〗 ) (Using cos 2x = 1 – 2sin2 x) (Using sin2 x = 1 – cos2 x ) = lim┬(x→0) ( 2 (12 − cos2 x) )/( 1−〖 cos〗⁡𝑥 ) = lim┬(x→0) ( 2 (1 − cos x)(1 +〖 cos〗⁡〖𝑥)〗 )/( 1−〖 cos〗⁡𝑥 ) = lim┬(x→0) 2 (1 + cos x) Putting x = 0 = 2 (1 + cos 0) = 2 (1 + 1) = 2 × 2 = 4

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo