Quotient of monotone functions is monotone? Suppose $f,g$ are monotone( say increasing) and differentiable and nonnegative. Both go from $\mathbb{R}^{\geq 1} \to \mathbb{R} $

Is $\frac{f}{g}$ also monotone ?
$f(x)=2x+\sin x,\ g(x)=x,\ 0\lt x\lt\infty$
Counterexample:
Let $f(x) = e^x+5, g(x) = x$. 
$$\frac{f}{g}(x) = \frac{e^x+5}{x}$$

Can you tell me why $\displaystyle{\frac{f}{g}}$ is not necessarily monotone?