Limit problem involving factorials and trigonometry

Question

The limit is: $$\lim_{n\to +\infty}\frac{n ln(\frac{((n+1)!)^{2n}}{n^{2n}(n!)^{2n}})+\frac{arctan(n)}{n}+\frac{sin(n)}{n}}{(-1)^n\sqrt{n}-4n}$$ After some calculations for the numerator i got to this conclusion: $$\lim_{n\to +\infty}\frac{2n}{(-1)^n\sqrt{n}-4n}$$ At this point it is difficult for me to work around $$(-1)^n$$ which is giving me some problems. Any hints? $$ $$ **I though about simplifying the n's which would lead to:$$\lim_{n\to +\infty}\frac{2}{\frac{(-1)^n}{\sqrt{n}}-4}$$

Answer 1

Note that $$\lim_{n\to +\infty}\frac{2n}{(-1)^n\sqrt{n}-4n} = \lim_{n\to +\infty}\frac{2}{\frac{(-1)^n}{\sqrt{n}}-4}$$

where $\frac{(-1)^n}{\sqrt{n}} \longrightarrow 0$.