Monday, 28 October 2019

real analysis - Verification of limnrightarrowinftysqrt[n]n3=1




I am interested in the limit




lim




Can we simply conclude that:
\lim_{n \rightarrow \infty} (\sqrt[n]{n})^3= 1^3=1.

I have proven that \sqrt[n]{n}\rightarrow1 earlier in this textbook. Also since the limit of a power is the power of the limit.


Answer



Yes, we can simply do that. Since the exponent ^3 is a constant neutral number (meaning we may interpret it as a fixed number of multiplications) we can move the limit inside of it. So if you already know \lim_{n\to\infty}\sqrt[n]n=1 then that's a full proof.


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