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GATE 2022 MA – Question 38

Real Analysis · Measure and Integration · 2 marks · Multiple choice

Continuous f:[−π,π]→ℝ satisfies f(x)>f(0)/2 for |x|<δ,0<δ<π. Put P_(n,δ)(x)=(1+cos x−cosδ)ⁿ. Which integral necessarily tends to zero as n→∞?

  1. ∫₀^(2δ) fP_(n,δ)dx
  2. ∫₋₂δ⁰ fP_(n,δ)dx
  3. ∫₋δ^δ fP_(n,δ)dx
  4. ∫_[−π,π]∖[−δ,δ] fP_(n,δ)dx

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Correct answer: (D) ∫_[−π,π]∖[−δ,δ] fP_(n,δ)dx

Explanation

Outside [−δ,δ], −1<−cosδ≤1+cos x−cosδ<1 except boundary points of measure zero, so P tends pointwise to zero and |P|≤1. Since f is bounded on the compact interval, dominated convergence proves **D**. Inside a neighborhood of zero the base exceeds 1 and f(0)>0 (by the supplied inequality at zero), so the other limits cannot generally be zero. If 2δ>π their displayed endpoints also exceed the given domain; D remains well defined.

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