I can prove that $\frac{\cos x}{x}$ is an infinitesimal for $x \to \infty$ with the squeeze theorem.
But trying to find the order of infinitesimal, I'm not sure if my reasoning is valid.
Here's what I thought:
We can show that the limit $$\lim_{x \to \infty} \frac{\cos x}{x} x^\alpha$$ Is zero for every $\alpha \lt 1$, while it's infinite for $\alpha \gt 1$. From considerations about the comparison between infinitesimals, we can easily infere that the order of infinitesimal of the function is thus $\alpha = 1$.
However, I'm aware that the limit with $\alpha = 1$ doesn't exist, so I'm not sure if my proof is applicable.
Could anyone shed light on my doubt?