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Calculus Level 4

The sequence of functions defined by

f 0 : x ↦ x f_0: x \mapsto \sqrt{x}

f n + 1 : x ↦ x + x − f n f_{n+1}: x \mapsto \sqrt{x+\sqrt{x-f_n}}

converges absolutely in the valid domain and codomain as n → ∞ n \to \infty

The domain and codomain of the sequence of functions are both R + \mathbb{R}^+

Which function does it converge towards?

Bonus : Determine the domain and range of f ∞ f_\infty

f : x ↦ 1 − 4 x − 3 2 f: x \mapsto \frac{1-\sqrt{4x-3}}{2} f : x ↦ − 1 + 4 x + 1 2 f: x \mapsto \frac{-1+\sqrt{4x+1}}{2} f : x ↦ 1 + 4 x − 3 2 f: x \mapsto \frac{1+\sqrt{4x-3}}{2} f : x ↦ − 1 − 4 x + 1 2 f: x \mapsto \frac{-1-\sqrt{4x+1}}{2}

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Jack Lam by Jack Lam , 22, Australia

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1 solution

  1. As n → ∞ n \rightarrow \infty , f n + 1 = f n = f ∞ f_{n+1} = f_n = f_\infty
  2. Use this in the recursive relationship.
  3. Evaluate at x = 1 x = 1 , You get f ∞ 2 = 1 + 1 − f ∞ f_\infty^2 = 1 + \sqrt{1-f_\infty}
  4. Evaluate the given options at x = 1 x = 1 , only C fits.
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