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Part BCSIR NET June 2025the-limit-must-satisfy-the-fixed-point-equation

The limit must satisfy the fixed point equation

Let f:Rf : \mathbb{R} \ Q→R\mathbb{Q} \to \mathbb{R} \ Q\mathbb{Q} be the function defined as f(x) = (3x + 2)/(4x + 3). Let x1∈Rx_{1} \in \mathbb{R} \ Q\mathbb{Q}. For n ≥ 1, define xnx_{n}₊1=f(xn)_{1} = f(x_{n}). Suppose that the sequence (xn)n≥1(x_{n})_{n}\ge_{1} converges to a real number ℓ\ell. Which of the following statements is true?

  1. A.If ℓ\ell is positive, then ℓ=3/2\ell = \sqrt{3}/2.
  2. B.If ℓ\ell is positive, then ℓ=1/2\ell = 1/\sqrt{2}.✓ Correct
  3. C.If ℓ\ell is negative, then ℓ=−3/2\ell = -\sqrt{3}/2.
  4. D.If ℓ\ell is negative, then ℓ=−1/2\ell = -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, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

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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