Does $x\cos(x)$ have oblique asymptotes?

Question

Looking at the graph of $x\cos(x)$ or $x\sin(x)$ etc., it looks like the magnitude of the waves are following a line. Are they oblique asymptotes or something else?

I am familiar with finding the oblique asymptotes of a rational function like $\frac{P(x)}{Q(x)}$ by dividing $Q(x)$ into $P(x)$; however, it doesn't seem like I can do that with $x \cos(x)$ et al. what is going on here? and how would you find the equation of the function that is 'controlling' the original function's behavior?

Answer 1

For a line to count as an asymptote, the distance between that line and the graph of the function has to tend to zero. This doesn't hold in your examples, so the lines are not asymptotes.

Answer 2

Hint: If you are to have an oblique asymptote, then you need a linear function $y = ax+b$ such that $\lim_{x\rightarrow\infty} \frac{x\sin x}{ax+b}$ exists. Can such a linear function exist?