Find the result of the $\lim_{n\to+\infty}\frac{\sum_{k=1}^{n}\frac{1}{k}}{\ln n}$

Question

Does there exist an alternative way to demonstrate and evaluate the result of the following limit without using an asymptotic expansion? $$\lim_{n\rightarrow+\infty}\dfrac{\displaystyle\sum_{k=1}^{n}\dfrac{1}{k}}{\ln n}$$ Using a spreadsheet presumably the result of the limit is $1$, and watching other posts like this and this related about Euler-Mascheroni constant $\gamma$, the result is obtained immediately. $${\displaystyle \lim_{n\rightarrow+\infty}\dfrac{\displaystyle\sum_{k=1}^{n}\dfrac{1}{k}}{\ln n}=\lim_{n\rightarrow+\infty}\dfrac{\ln n+\gamma+O\begin{pmatrix}\dfrac{1}{n}\end{pmatrix}}{\ln n}}=1$$ Is that correct? However, I would like to know if there is other way to calculate that result using calculus in a context of firsts courses of engineering careers. Maybe with convergence criteria for series?