a limit containing a digamma function

Question

I wonder if the following limit is correct and how to prove it $$ \lim _{x \rightarrow+\infty} \frac{4}{p^{2}} x^{4}\left(\frac{1}{x^{2}}+\frac{2}{x}(\log 2+\Psi(x))+\Psi^{\prime}(x)\right)>0 $$ where $p>0$

Answer 1

It is sufficient to use the asymptotic expansions you mentioned in the comments. From the asymptotic expansions it follows that as $x\to\infty$ $$ \frac{4}{p^{2}} x^{4}\left(\frac{1}{x^{2}}+\frac{2}{x}(\log 2+\Psi(x))+\Psi^{\prime}(x)\right)\sim\frac{4}{p^2}x^3(1+\log 4+2\log x)+\mathcal O(x^2)\to\infty. $$ And so for all $p>0$ the limit will diverge to $\infty$ and thus is greater than zero.