\lim_{n\to \infty } \, \frac{2^{n+1}+3^{n+1}}{2^n+3^n}=\lim_{n\to \infty } \, \frac{2*2^n+3*3^n}{2^n+3^n}=\lim_{n\to \infty } \, \frac{2^n+3^n+2^n+3^n+3^n}{2^n+3^n}=\lim_{n\to \infty } \left(1+1+\frac{3^n}{2^n+3^n}\right)=\lim_{n\to \infty } \left(2+\frac{3^n}{3^n\left(\left(\frac{2}{3}\right)^n+1\right.}\right)=\lim_{n\to \infty } \left(2+\frac{1}{\left(\left(\frac{2}{3}\right)^n+1\right.}\right)=2+\frac{1}{0+1}=3.