# Proving $\lim_{x\to\infty}\frac{\sqrt x\cos(x-x^2)}{x+1} = 0.$

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 $0$.

• Based on intuition, the $\cos$ part doesn't play any role in how the magnitude changes, so it is basically just between $\sqrt{x}$ and $x$, which one of them grows more. – IllidanS4 Mar 27 '18 at 20:46

Note that

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

and

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

Let $-1\le \cos \left(x-x^2\right)\le 1$, then $$\lim _{x\to \infty }\left(\frac{\sqrt{x}\left(-1\right)}{x+1}\right)\le \lim _{x\to \infty }\left(\frac{\sqrt{x}\cos \left(x-x^2\right)}{x+1}\right)\le \lim _{x\to \infty }\left(\frac{\sqrt{x}\cdot1}{x+1}\right)$$ It follows $$\lim _{x\to \infty }\left(\frac{\sqrt{x}\left(\pm1\right)}{x+1}\right) = \pm\lim _{x\to \infty }\left(\frac{\sqrt{x}}{x+1}\right) = \pm\lim _{x\to \infty }\left(\frac{\frac{1}{\sqrt{x}}}{1+\frac{1}{x}}\right)=\pm\frac{0}{1} = 0$$

@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 $1/{\sqrt x}$.