asymptotic sequence

I am asked to prove that $x^n(a+\cos(x^{-n})$ is an asymptotic sequence for $n=0,1,2,...$, $a>1$, $x\rightarrow 0$ but its derivative wrt x isn't an asymptotic sequence.

https://www.encyclopediaofmath.org/index.php/Asymptotic_sequence

The first part is easy, I failed at proving the second part.

I have the next limit to calculate, I believe this sequence doesn't converge at all, but how to find two converging to 0 sequences such we get to different limits?

I want to show that:

$$\lim_{x\rightarrow 0}\frac{(n+1)x^n(a+\cos(x^{-n-1}))+\frac{n+1}{x}\sin(x^{-n-1})}{nx^{n-1}(a+\cos(x^{-n}))+\frac{n}{x}\sin(x^{-n})}$$

doesn't exist or doesn't equal zero, I think that it doesn't exist.

Any hints or solutions?

Thanks.

-

It's a bit cleaner if you multiply both numerator and denominator by $x$. Try the sequence $x_k=(\frac{\pi}{2}+2\pi k)^{-1/(n+1)}$: it achieves $\sin x_k^{-n-1}=1$ in the numerator, which makes the numerator bounded from below. And since the denominator is bounded from above, the ratio cannot converge to zero.