calculus - Proving $lim_{xtoinfty}frac{sqrt xcos(x-x^2)}{x+1} = 0.$

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 $0$.

Answer

Note that

$$-\frac{\sqrt x}{x+1}\le \frac{\sqrt x\cos(x-x^2)}{x+1}\le\frac{\sqrt x}{x+1}$$

and

$$\frac{\sqrt x}{x+1}\to 0$$