Ex 13.1, 21 - Evaluate: lim x->0 (cosec x - cot x) - Class 11 - Limits - Of Trignometric functions

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  1. Chapter 13 Class 11 Limits and Derivatives
  2. Serial order wise
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Ex 13.1, 21 (Method 1) Evaluate the Given limit: lim┬(x→0) (cosec x – cot x) lim┬(x→0) (cosec x – cot x) = lim┬(x→0) (1/sin⁡𝑥 −cos⁡𝑥/sin⁡𝑥 ) = lim┬(x→0) (1 −〖 cos〗⁡𝑥)/sin⁡〖𝑥 〗 Putting x = 0 = (1 − cos⁡〖0 〗)/sin⁡0 = (1 − 1)/0 = 0/0 Hence, we need to simplify lim┬(x→0) (1 −〖 cos〗⁡𝑥)/sin⁡〖𝑥 〗 = lim┬(x→0) (1 −〖 cos〗⁡𝑥)/sin⁡〖𝑥 〗 × (𝟏 + 𝐜𝐨𝐬⁡𝒙)/(𝟏 + 𝒄𝒐𝒔⁡𝒙 ) = lim┬(x→0) (1^2 −〖 cos^2〗⁡𝑥)/(sin⁡〖𝑥 〗 (1 + cos⁡𝑥 ) ) = lim┬(x→0) (𝟏 −〖 〖𝒄𝒐𝒔〗^𝟐〗⁡𝒙)/(sin⁡〖𝑥 〗 (1 + cos⁡𝑥 ) ) = lim┬(x→0) 〖 〖𝐬𝐢𝐧〗^𝟐〗⁡𝒙/(sin⁡〖𝑥 〗 (1 + cos⁡𝑥 ) ) = lim┬(x→0) 〖 𝑠𝑖𝑛〗⁡𝑥/((1 + cos⁡𝑥 ) ) Putting x = 0 = 〖 𝑠𝑖𝑛〗⁡0/((1 + cos⁡0 ) ) = 0/(1 + 1) = 0/2 = 0

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