Let $lim_{n to infty} n * x_n =a$. Why it follows that

$$lim_{n to infty} (n *(1-x_n)^{1-frac{k}{n}}-n)=-a$$

where $kin mathbb{N}$.

I see that $1-frac{k}{n} to 1$ and then $n-x_n*n to n-a$ and hence the whole term converges to $ ;-a$. But this is not a permissible mathematical argumentation.