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.