I need to prove that, it's clearly obvius, but i dont know how to prove that:
$$ \lim_{n\to \infty}n^\frac 1n = 1 $$
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Note that $n^{1/n} \geq 1$ for $n\geq 1$ so there is a sequence of non-negative reals $\delta(n)$ s.t. $n^{1/n} = 1 + \delta (n)$ for each $n$. Show that $\delta(n) \to 0$ as $n\to \infty$ by raising both sides to the power of $n$.
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Explicit solution is given in the other answers.
But, it is worth noting that the discrete version of L'Hospital theorem is Stolz-Cesaro theorem (with explicit solution here or here).
In it, take $a_n = \ln n$ and $b_n=n$. Check its conditions and apply it. This will give you $$\lim_{n\to \infty} \frac{\ln n}{n} = 0.$$ Imply the limit you need from this.
