# reason for the limit of a sequence

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.