Example 4 - Chapter 13 Class 11 Limits and Derivatives

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Example, 4 Evaluate: (i) lim﷮x→0﷯ sin﷮4x﷯﷮sin 2x﷯ lim﷮x→0﷯ sin﷮4x﷯﷮sin 2x﷯ = lim﷮x→0﷯ sin 4x × lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ Multiplying & dividing by 4x = lim﷮x→0﷯ sin 4x . 4𝑥﷮4𝑥﷯ × lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ = lim﷮x→0﷯ sin﷮4𝑥﷯﷮4𝑥﷯ . 4x × lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ = 𝐥𝐢𝐦﷮𝐱→𝟎﷯ 𝒔𝒊𝒏﷮𝟒𝒙﷯﷮𝟒𝒙﷯ × lim﷮x→0﷯ 4x × lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ = 1 × lim﷮x→0﷯ 4x × lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ = 1 × lim﷮x→0﷯ 4x × lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ = lim﷮x→0﷯ 1﷮ sin﷮2𝑥﷯﷯ . 4x Multiplying & dividing by 2x = lim﷮x→0﷯ 1﷮sin 2x﷯ × 4x × 2𝑥﷮2𝑥﷯ = lim﷮x→0﷯ 2𝑥﷮sin 2x﷯ × 4𝑥﷮2𝑥﷯ = lim﷮x→0﷯ 2𝑥﷮ sin﷮2𝑥﷯﷯ × 2 = 2 lim﷮x→0﷯ 2𝑥﷮ sin﷮2𝑥﷯﷯ = 2 ÷ 𝐥𝐢𝐦﷮𝐱→𝟎﷯ 𝐬𝐢𝐧﷮𝟐𝒙﷯﷮𝟐𝒙﷯ = 2 ÷ 1 = 2﷮1﷯ = 2 Example 4 Evaluate: (ii) lim﷮x→0﷯ tan﷮x﷯﷮x﷯ lim﷮x→0﷯ tan﷮x﷯﷮x﷯ = lim﷮x→0﷯ 1﷮𝑥﷯ × tan x = lim﷮x→0﷯ 1﷮𝑥﷯ × sin﷮x﷯﷮ cos﷮x﷯﷯ = lim﷮x→0﷯ sin﷮x﷯﷮x﷯ × 1﷮ cos﷮𝑥﷯﷯ = 𝐥𝐢𝐦﷮𝐱→𝟎﷯ 𝐬𝐢𝐧﷮𝐱﷯﷮𝐱﷯ × lim﷮x→0﷯ 1﷮cos x﷯ = 1 × lim﷮x→0﷯ 1﷮cos x﷯ = 1 × lim﷮x→0﷯ 1﷮cos x﷯ = lim﷮x→0﷯ 1﷮cos x﷯ Putting x = 0 = 1﷮ cos﷮0﷯﷯ = 1﷮1﷯ = 1