Ex 13.1, 17 - Evaluate: lim x->0 cos2x - 1 / cosx - 1 - Ex 13.1

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  1. Chapter 13 Class 11 Limits and Derivatives
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Ex13.1, 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⁡〖𝟐𝒙 〗) − 1)/cos⁡〖𝑥 − 1〗 = lim┬(x→0) (1 − 2 sin2 x − 1 )/cos⁡〖x − 1〗 = lim┬(x→0) ( − 2 𝐬𝐢𝐧𝟐 𝐱 )/cos⁡〖x − 1〗 = lim┬(x→0) ( − 2 (𝟏 − 𝐜𝐨𝐬𝟐 𝐱) )/cos⁡〖x − 1〗 = lim┬(x→0) ( − 2 (1 − cos2 x) )/(−1 (1−〖 cos〗⁡〖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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