Is it alright to substitute 0 for 1/n in this limit problem?

Evaluate: $\lim_{n \to \infty}\displaystyle\sum_{r=1}^{n} \dfrac{1}{2n + 2r-1}$

To solve this I used the following approach: $S = \lim_{n \to \infty}\displaystyle\sum_{r=1}^{n} \dfrac{\frac 1n}{2 + 2\frac rn-\frac1n}$

$= \lim_{n \to \infty}\displaystyle\sum_{r=1}^{n} \dfrac{\frac 1n}{2 + 2\frac rn-\color{red}{0}} = \dfrac 12\int _0^1 \dfrac{1}{1+x}\mathrm{dx} = \ln(\sqrt 2)$

Though the answer is correct, I am unsure about my second step in which I have replaced $1/n$ by $0$. Is that allowed? I have tried it in some other problems too and it works.

from Hot Weekly Questions - Mathematics Stack Exchange