Calculating $\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 $ I want to calculate the following limit:
$\displaystyle\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 $
I tried the following:
$\displaystyle\lim_{x\rightarrow +\infty} x \sin(x) + \cos(x) - x^2 = $
$\displaystyle\lim_{x\rightarrow +\infty} x^2 \left( \frac{\sin(x)}{x} + \frac{\cos(x)}{x^2} - 1 \right) = +\infty$
because 
$\displaystyle\lim_{x\rightarrow +\infty} x^2 = +\infty$,
$\displaystyle\lim_{x\rightarrow +\infty} \frac{\sin(x)}{x} = 0$, and
$\displaystyle\lim_{x\rightarrow +\infty} \frac{\cos(x)}{x^2} = 0$,
but I know from the plot of the function that this limit goes to $- \infty$, so I'm clearly doing something wrong. Sorry in advance for the simple question and perhaps for some silly mistake.

Hint: $-\frac{1}{2}x^2\geq x+1-x^2\geq (x\sin x+\cos x-x^2)$ for $x\,$ large enough. Now let $x$ go to infinity.
