Find $\lim_{n\to\infty} \sqrt[n]{n}$

Question

Find: $$\lim_{n\to\infty} \sqrt[n]{n}$$

How does one not having any intuition able to do this?

Answer 1

Note that $n^{\frac{1}{n}} = e^{\frac{1}{n}\log(n)}$. Since $\frac{1}{n}\log(n)\to 0$ as $n\to\infty$ and since the exponential function is continuous, we have $\lim_{n\to\infty} n^{\frac{1}{n}} = \lim_{n\to\infty} e^{\frac{1}{n}\log(n)} = e^0 = 1$.

Answer 2

Hints (for you to work out and justify):

$$\forall\,n\in\Bbb N\;,\;\;\sqrt[n]n\ge 1\implies \sqrt[n]n=1+c_n\;,\;\;c_n\ge 0\implies$$

$$n=(1+c_n)^n=\sum_{k=0}^n\binom{n}{k}c_n^k\ge \frac{n(n-1)}{2}c_n^2\implies c_n\le \sqrt\frac{2}{n-1}\implies c_n\xrightarrow [n\to\infty]{}0\implies$$

$$\sqrt[n]n\xrightarrow[n\to\infty]{}1$$