The limit of limits!

Calculus Level 5

Given \(f(x)\) is a polynomial of \(n\)th degree and

\[\large \lim_{x \to 0} \frac{2^{r-1}(x-2)f(x)-x^{r}+2^{r}}{x^{r}(x-2)}=\frac{1}{2} \quad \forall r \in \mathbb N \cap [1,n] \]

Find \(\displaystyle \lim_{n \to \infty}f(1)\).


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