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JEE Main 2017

JEE Main 2017 Mathematics question on Limits Continuity and Differentiability

lim⁡x→π/2cot⁡x−cos⁡x(π−2x)3\lim_{x \to \pi/2} \dfrac{\cot x - \cos x}{(\pi - 2x)^3} equals
  1. A.
    124\dfrac{1}{24}
  2. B.
    116\dfrac{1}{16}
  3. C.
    18\dfrac{1}{8}
  4. D.
    14\dfrac{1}{4}
Show answer and solution

Correct option: B

Solution

Let t = π/2 - x, so as x → π/2, t → 0.

cot x = cot(π/2 - t) = tan t, cos x = cos(π/2 - t) = sin t, and (π - 2x)³ = (2t)³ = 8t³.

The limit becomes: (tan t - sin t)/(8t³) = (sin t/cos t - sin t)/(8t³) = sin t(1 - cos t)/(8t³ cos t).

As t → 0: sin t/t → 1, (1 - cos t)/t² → 1/2, cos t → 1.

So the limit = (1)(1/2)/(8·1) = 1/16.

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