Proving that lim n → ∞ x 1 / n = 1 at x &gt; 0

Answered: "Proving that lim n → ∞ x 1..."

Damon Stokes

2022-06-17

Proving that

lim_{n \to \infty} x^{1 / n} = 1

x > 0

zalitiaf

Beginner2022-06-18Added 27 answers

An easier approach, if permitted, is to let

y = \log (x)

Then, regardless of whether y is positive, negative, or zero, you have that

lim_{n \to \infty} \frac{y}{n} = 0.

Therefore, it is routine to set up an

ε

demonstration, choosing N large enough so that for all

n \geq N

, you have that

e^{(y / n)}

is in the neighborhood of

(1 - ε, 1 + ε) .

That is, you want to choose N large enough so that

| \frac{y}{N} | < \log (1 + ε)

and

| \frac{y}{N} | < | \log (1 - ε) | .

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