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Part CCSIR NET December 2025a-strictly-positive-integrand-over-a-positive-length-interval-can-approach-0-as-x-shrinks-but-never-equal-it-for-x-greater-than-0

A strictly positive integrand over a positive length interval can approach 0 as x shrinks but never equal it for x greater than 0

Let f:(0,∞)→(0,∞)f : (0,\infty) \to (0,\infty) be the function defined by f(x)=∫0f(x) = \int_{0}ˣ t/(1+t2)\sqrt{t}/(1+t^{2}) dt, where t\sqrt{t} denotes the positive square root for t>0. Which of the following statements are true?

  1. A.f is a uniformly continuous function.✓
  2. B.f is a bounded function.✓
  3. C.There exists x∈(0,∞)x\in(0,\infty) such that f(x)=0.
  4. D.The derivative of f is continuous.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: If ∫₀^∞ f converges then f(x) → 0

More on this topic

The chapter behind this: Improper integrals — thresholds and tests — free to read

From Integration › Improper integrals and convergence tests

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