Wednesday, 13 September 2017

limits - Evaluation limntoinftyfraclogknnepsilon




Evaluate where ϵ>0,k are constants



\lim_{n\to \infty}\frac{{\log^k n}}{n^{\epsilon}}





L'Hopital can't help here, also I tried to use \log rules but it didn't helped, I know that \log grows slower then polynom, but n^\epsilon is not polynom, how can I evaluate this limit? thank you


Answer



Write
\frac{(\log n)^k}{n^\varepsilon}= \left(\frac{\log n}{n^{\varepsilon/k}}\right)^{\!k}
For r>0, we have

\lim_{x\to\infty}\frac{\log x}{x^r}= \lim_{x\to\infty}\frac{1/x}{rx^{r-1}}= \lim_{x\to\infty}\frac{1}{rx^r}=0


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