I have these two functions over real $x>0$ (with $a,b$ finite real constants) $$ g(x,a)=\frac a x \cos x+x-15\sin x \qquad (1) $$ $$ g(x,b)=\frac b x \cos x+x-15\sin x \qquad (2)$$
I want to evaluate these functions as $x\to\infty\;$; am I correct that in this limit, the functions $(1)$ and $(2)$ behave in a similar way (independent of $a$ and $b$) as follows? $$ g(x,a)\approx g(x,b)\approx g(x)\approx x-15\sin x+\mathcal{O}(\frac 1x) $$
Asymptotically, if you have a sum of bounded and unbounded monotonic terms, you can ignore all the bounded terms. You have $|15 \sin x|\leq 15$, so adding or subtracting a number in this range is going to have less and less an impact on something that is growing to infinity, with the impact in ratio terms going to 0.