A problem with limit How to attack this one?
Does the following limit exists:
$$\lim_{x\to +\infty}\dfrac {\cos^5x\sin^5x} {x^8\sin^2x-2x^7\sin x\cos^2x+x^6\cos^4x+x^2\cos^8x}$$

The denominator can be rewritten as the sum of the two nonnegative terms
$$
(x \sin x - \cos^2 x)^2 x^6 + x^2 \cos^8 x.
$$
For all integral $n\ge 1$, when $x$ is in the interval $[2\pi n-\frac{\pi}{4}, 2\pi n+\frac{\pi}{4}]$ or $[2\pi n+\frac{3\pi}{4}, 2\pi n+\frac{5\pi}{4}]$, $|\cos x|\ge 2^{-1/2}$, so the second term will be at least $x^2/16$.  On the other hand, if $x$ is in the interval $[2\pi n+\frac{\pi}{4}, 2\pi n+\frac{3\pi}{4}]$ or $[2\pi n+\frac{5\pi}{4}, 2\pi n+\frac{7\pi}{4}]$, $|x \sin x|\ge  |\sin x|\ge 2^{-1/2}$ and $|\cos^2 x|\le\frac 12$, so the first term will be at least $\frac 12 x^6 (1-2^{-1/2})^2$.  Assume that $x\ge 2\pi$.  Then $\frac 12 x^6 (1-2^{-1/2})^2\ge x^2/16$, so, regardless of the interval $x$ belongs to, the denominator is bounded below by $x^2/16$.  Therefore, the limit exists and is zero.
