Is it possible to define such an operator $\operatorname{\Gamma}$ that satisfies $ \lim_{n\to \infty} {\Gamma} (f(n))=\beta $?

Question

Let, the function $f:\mathbb N^+ \longrightarrow \mathbb R$ is given. Is it possible to define such an operator $\operatorname{\Gamma}$ that satisfies the following conditions:

For any $n\in\mathbb {N^+}$ we have

$$\operatorname{\Gamma} (f(n))=\alpha$$and

$$\displaystyle \lim_{n\to \infty} \operatorname{\Gamma} (f(n))=\beta $$

where $\alpha,\beta \in \mathbb R$ and $\alpha\neq \beta.$

Is this mathematically possible?

Answer 1

No, simply because $$\lim_{n\to\infty}\Gamma(f(n)) = \lim_{n\to\infty}\alpha = \alpha\neq\beta$$