Ex 12.1, 22 - Chapter 12 Class 11 Limits and Derivatives

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Ex 13.1, 22 - lim x->pi/2 tan 2x/x-pi/2 - Chapter 13 Class 11

Ex 13.1, 22 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Ex 13.1, 22 - Chapter 13 Class 11 Limits and Derivatives - Part 3

 

Transcript

Ex 12.1, 22 lim┬(x → π/2) tan⁡2x/(x − π/2) lim┬(x → π/2) tan⁡2x/(x − π/2) Putting y = x – π/2 When x → 𝜋/2 y → 𝜋/2 – 𝜋/2 y → 0 So, our equation becomes lim┬(x→π/2) tan⁡2x/(x − π/2) = lim┬(y→0) (tan⁡2(𝜋/2 + 𝑦)/𝑦) = lim┬(y→0) ((〖tan 〗⁡〖(𝜋 + 2𝑦〗))/𝑦) = lim┬(y→0) (tan⁡2𝑦/𝑦) = lim┬(y→0) (1/𝑦 . sin⁡2𝑦/cos⁡2𝑦 ) = lim┬(y→0) (sin⁡2𝑦/𝑦 . 1/cos⁡2𝑦 ) = lim┬(y→0) sin⁡2𝑦/𝑦 ×lim┬(y→0) 1/cos⁡2𝑦 Multiply & Divide by 2 (As tan⁡〖(𝜋+𝑥〗)=tan x) = lim┬(y→0) (sin⁡2𝑦/𝑦 "× " 2/2).lim┬(y→0) 1/cos⁡2𝑦 = 2 lim┬(y→0) (𝒔𝒊𝒏⁡𝟐𝒚/𝟐𝒚).lim┬(y→0) 1/cos⁡2𝑦 Using ( lim)┬(x→0) (sin⁡x )/x = 1 Replacing x by 2y. lim┬(x→0) sin⁡2𝑦/2y = 1 = 2 × 1 × lim┬(y→0) 1/cos⁡2𝑦 = 2 × 1/cos⁡〖2(0)〗 = 2/cos⁡0 = 2/1 = 2 (As cos 0 = 1)

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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, Social Science, Physics, Chemistry, Computer Science at Teachoo.