Let a real sequence {$x_n$}$_{n\ge 1}$ be such that $\lim_{n \to \infty} nx_n=0$
Find all real values of t such that $\lim_{n \to \infty} x_n(\log n)^t=0$
My approach is $\log n\to \infty$, so $t > 0$, but I am not sure about it. Also, I am not sure how to write the solution.
Since $\;\lim\limits_{n\to\infty}\cfrac{(\log n)^t}{n}=0\;$ for all $\;t\in\mathbb{R}\;,\;$
$\lim\limits_{n\to\infty}x_n\left(\log n\right)^t=\lim\limits_{n\to\infty}nx_n\cfrac{\left(\log n\right)^t}{n}=$
$=\lim\limits_{n\to\infty}nx_n\cdot\lim\limits_{n\to\infty}\cfrac{(\log n)^t}{n}=0\;,\;$ for all $\;t\in\mathbb{R}\;.$
