TITLE: Proving $ \lim_{x\to\infty}\frac{\sqrt x\cos(x-x^2)}{x+1} = 0. $
QUESTION [14 upvotes]: I have to prove that 
$$ \lim_{x\to\infty}\frac{\sqrt x\cos(x-x^2)}{x+1} = 0. $$
I tried squaring both the denominator and numerator to get rid of $\sqrt{x}$ but then $\cos$ becomes $\cos^2$ and I do not know how to solve it/simplify it further. 
I have also tried to use the squeeze theorem with $a_n = 0$, and $$ b_n =\frac{\sqrt x\cos(x-x^2)}{x+1}, $$ but to no avail. I could not find another function that is greater than $b_n$ that has a limit of $0$.

REPLY [7 votes]: @gimusi shows 'why'.
And this is 'how':
The first thing to note is that cosine remains between $-1$ and $+1$, no matter what its argument is.
The second one is that a linear expression in the denominator grows faster than a square root in the numerator.
Then you can squeeze the cosine with $$\pm\frac{\sqrt x}{x+1} $$ whose absolute value is in turn less than $1/{\sqrt x}$.