So I have something like this: $\lim_{n \rightarrow \infty} (-2 n+\sqrt{2+5 n+4 n^2})$
On the lesson, it was solved by multiplying by the conjugate and therefore arriving at the polynomial/polynomial form. I understand that, but for my current knowledge of limitis, it stands in contradiction to the properties of limits which state that $\lim_{n \rightarrow \infty}(a_n+b_n)=\lim_{n \rightarrow \infty}a_n+\lim_{n \rightarrow \infty}b_n$ and that $\lim_{n \rightarrow \infty}\sqrt{a_n}^k = \sqrt{\lim_{n \rightarrow \infty}a_n}^k$.
Why can't we use these two here? If we could, wouldn't it become $\lim_{n \rightarrow \infty}-2n=-\infty$ and $\lim_{n \rightarrow \infty}\sqrt{2+5 n+4 n^2}=\sqrt{\lim_{n \rightarrow \infty}2+5 n+4 n^2}=\infty$ and then $\infty-\infty=0$? Why would such reasoning be wrong?