Correctness of limit calculation/correctness check $\lim_\limits{n\to\infty}(n!)^{1/n^2}=1$ , $\;\lim_\limits{n\to\infty}n\!\cdot\!\ln(1+1/n)\,$.

Question

Correctness of limit calculation/correctness check $\lim_\limits{n\to\infty}\big(n!\big)^{\frac1{n^2}}=1$ , $\;\lim_\limits{n\to\infty}n\!\cdot\!\ln\left(1+\dfrac1n\right)\,$.

Calculated Simple Limits. I am wondering if my reasoning for the calculation is correct. The limits in the square are known limits that I don't need to prove.

Answer 1

I would recommend the following approach: \begin{align*} 1\leq (n!)^{1/n^{2}} \leq (n^{n})^{1/n^{2}} = n^{\frac{1}{n}} = \exp\left(\frac{\ln(n)}{n}\right) \xrightarrow{n\to\infty} 1 \end{align*}

Now you can apply the squeeze theorem.

Hopefully this helps!