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

Ex 13.1, 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﷯﷯ Divide & Multiply by 2y = lim﷮y→0﷯ sin﷮2𝑦﷯﷮𝑦﷯× 2𝑦﷮2𝑦﷯﷯. lim﷮y→0﷯ 1﷮ cos﷮2𝑦﷯﷯ = lim﷮y→0﷯ sin﷮2𝑦﷯﷮2𝑦﷯× 2𝑦﷮𝑦﷯﷯. lim﷮y→0﷯ 1﷮ cos﷮2𝑦﷯﷯ = lim﷮y→0﷯ sin﷮2𝑦﷯﷮2𝑦﷯× 2﷯. lim﷮y→0﷯ 1﷮ cos﷮2𝑦﷯﷯ = 2 𝐥𝐢𝐦﷮𝐲→𝟎﷯ 𝒔𝒊𝒏﷮𝟐𝒚﷯﷮𝟐𝒚﷯﷯. lim﷮y→0﷯ 1﷮ cos﷮2𝑦﷯﷯ = 2 . 1 . lim﷮y→0﷯ 1﷮ cos﷮2𝑦﷯﷯ = 2 . lim﷮y→0﷯ 1﷮ cos﷮2𝑦﷯﷯ = 2. 1﷮ cos﷮2(0)﷯﷯ = 2﷮ cos﷮0﷯﷯ = 2﷮1﷯ = 2