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BITSAT2018MathematicsContinuity and DifferentiabilityActual

Let f x = x p sin 1 x , x ≠ 0 0 , x = 0 then f x is continuous but not differentiable at x = 0 , if

Options

  1. Ap < 0
  2. Bp = 0
  3. C0 < p ≤ 1
  4. Dp ≥ 1

Correct answer

C. 0 < p ≤ 1

Step-by-step solution

We have, f x = x p sin 1 x , x ≠ 0 0 ,                     x = 0 For f x to be continuous at x = 0 lim x → 0 f x = f 0 ⇒ lim x → 0 x p sin 1 x = 0 ⇒ lim x → 0 x p × lim x → 0 sin 1 x = 0 ⇒ lim x → 0 x p × value between  - 1  to   1 = 0 ∴ p > 0 Now, RHD at RHD at  x = 0 = lim h → 0 h p sin 1 h - 0 h = lim h → 0 h p - 1 sin 1 h LHD at  x = 0 =

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