$a_n=\frac{1000^{\sqrt[100]{n}}}{(1,01)^n}+n^{1000}(0,99)^{\sqrt[100]{n}}$
Is it correct to write that $\lim \limits_{n \to \infty\ }a_n=\lim \limits_{n \to \infty\ } \frac{1000^{\sqrt[100]{n}}}{(1,01)^n} + \lim \limits_{n \to \infty\ } n^{1000}(0,99)^{\sqrt[100]{n}}=...=0+0=0$?
The limit function is a linear function, so yes, it's okay and acceptable. You have to show explicitly how the limit of the two terms is $0$ though, to make sure you have a rigorous approach.