I was trying to solve $$\lim _{n\to \infty }\left(\frac{n3^n\left(\sqrt{n+1}-\sqrt{n}\right)}{3^n\sqrt{n+1}+2^n}\right)$$
What I have tried is dividing the numerator and denominator by $3^n$ and then multiplying by the conjugate of the numerator but I don't obtain the answer which I think is $\frac{1}{2}$.
What you suggest is correct. We have $$ \ldots = \frac{n (n+1 - 1)}{\sqrt{n+1} (\sqrt{n+1} + \sqrt{n}) + (\frac{2}{3})^n (\sqrt{n+1} + \sqrt{n})} = \frac{n}{n+1 + \sqrt{n^2 + n} + \text{small}} = \ldots $$