<s>[INST] Limit involving the Sine integral function $$
\mbox{Prove that}\qquad
\lim_{x \to \infty}\left[\vphantom{\large A}%
x\,\mathrm{si}\left(x\right)+ \cos\left(x\right)\right]
= 0
$$
where we define 
$$\mathrm{si}\left(x\right) =
- \int^{\infty}_{x}\frac{\sin\left(t\right)}{t}\,\mathrm{d}t
$$
I have no clue how to start. I have verified the result using wolframalpha.
 [/INST] Define $$Si(x) = \int^x_0 \frac{sin t}{t} dt$$
By the relation $Si(x) = \frac{\pi}{2} + si(x) $
And the asymptotic series expansion of $si(x)$:
$$si(x) = -\frac{\cos x}{x} (1- \frac{2!}{x^2} +\frac{4!}{x^4} -O(1/x^6))-\frac{\sin x}{x} (\frac{1}{x}- \frac{3!}{x^3} +\frac{5!}{x^5} -O(1/x^7)) $$
Therefore
$$xsi(x)+\cos x = -\cos x (1- \frac{2!}{x^2} +\frac{4!}{x^4} -O(1/x^6))-\sin x (\frac{1}{x}- \frac{3!}{x^3} +\frac{5!}{x^5} -O(1/x^7))+\cos x \\
=   -\cos x (- \frac{2!}{x^2} +\frac{4!}{x^4} -O(1/x^6))-\sin x (\frac{1}{x}- \frac{3!}{x^3} +\frac{5!}{x^5} -O(1/x^7))$$
When x goes up to infinity, $\cos x$ falter between $[-1,1]$ but $x^n$ rise to infinity  so $1/x^n$ goes to 0 where $n \geq 0$ is a natural number. 
</s>