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Part BCSIR NET December 2025rationalise-the-radical-difference-before-taking-the-limit

Rationalise the radical difference before taking the limit

For a non-negative real number a, let a\sqrt{a} denote its non-negative square-root. Consider the function f:R→Rf : \mathbb{R} \to \mathbb{R} given by f(x)=x(x2+3−x2+2)f(x) = x(\sqrt{x^{2}+3} - \sqrt{x^{2}+2}). Which of the following statements is true?

  1. A.lim⁡(x→∞)f(x)=∞\lim (x\to\infty) f(x) = \infty
  2. B.lim⁡(x→∞)f(x)=0\lim (x\to\infty) f(x) = 0
  3. C.lim⁡(x→∞)f(x)=1\lim (x\to\infty) f(x) = 1
  4. D.lim⁡(x→∞)f(x)=1/2\lim (x\to\infty) f(x) = 1/2✓

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: aₙ^{1/n} → L ⇒ aₙ₊₁/aₙ → L

More on this topic

The chapter behind this: Sequences in ℝ — the implication map — free to read

From The Real Line › Sequences: convergence, monotone, Bolzano–Weierstrass, Cauchy

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