Find out the answer to "Trying to solve:limx→∞ln1000xx5"

Jakobe Norton

Answered question

2022-04-10

Trying to solve:

lim_{x \to \infty} \frac{\ln^{1000} x}{x^{5}}

Terzago66cl

Beginner2022-04-11Added 8 answers

Be careful using limit operation.
First, let show that

lim_{x \to + \infty} \frac{\ln x}{x} = 0

. For

t \geq 1

, we have

t \geq \sqrt{t}

which imply for

x \geq 1

0 \leq \ln x = \int_{1}^{x} \frac{d t}{t} \leq \int_{1}^{x} \frac{d t}{\sqrt{t}} = 2 \sqrt{x} - 2 \leq 2 \sqrt{x}

Then, for any

a, b > 0

and

x > 1

, we have

\frac{\ln^{b} x}{x^{a}} = {(\frac{\ln x}{x^{\frac{a}{b}}})}^{b} = {(\frac{b}{a})}^{b} {(\frac{\ln (x^{\frac{a}{b}})}{x^{\frac{a}{b}}})}^{b}

which answer your question.

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