Limits practice question 16

The $\cos(2x)$ term keeps oscillating and never settles. How big can it ever be, next to $5x$ when $x$ is huge?

Divide the top and the bottom by $x$. What happens to $\dfrac{\cos(2x)}{x}$ as $x \to \infty$?

$\cos(2x)$ stays between $-1$ and $1$, so $\left|\dfrac{\cos(2x)}{x}\right| \leq \dfrac{1}{x}$ for $x > 0$.