Does More Surds Make A Difference?

Calculus Level 3

True or false : ⋯ x = 1 \sqrt{\sqrt{\sqrt{\cdots \sqrt{ x}}}} = 1

The infinitely nested radical equation above holds true for all x ≥ 0 x\geq 0 .

True False

Pi Han Goh by Pi Han Goh , 30, Malaysia

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3 solutions

Matteo Monzali
May 29, 2016

Just try with 0, in fact 0 = 0 \sqrt0=0 .

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Toze Gonçalves
Jun 6, 2016

I believe that 0 ∞ \sqrt[\infty]{0} is indeterminated and not 0 => lim ⁡ x → + ∞ 0 x = lim ⁡ x → + ∞ 0 1 x = 0 lim ⁡ x → + ∞ 1 x = 0 0 \lim_{x \to +\infty} \sqrt[x]{0} = \lim_{x \to +\infty} {0}^{\frac {1}{x}} = {0}^{ \lim_{x \to +\infty} \frac {1}{x}} = 0^{0} Am I wrong? some faulty reasoning?

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Nope. It's wrong. Yes it is lim ⁡ x → ∞ 0 1 / x \displaystyle \lim_{x\to\infty} 0^{1/x} . But that doesn't mean that the following steps follow. Read up indeterminate forms .

Pi Han Goh - 5 years ago
Akul Sharma
Jun 5, 2016

Infinite sqrts of any natural number except 0 is 1

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However, the domain in this question has also included 0.

Kobe Cheung - 5 years ago

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