Monotonicity of a fraction

Question

If I have a fraction $f(x) = \dfrac{n(x)}{d(x)}$, where $n(x)$ increases monotonically and $d(x)$ decreases monotonically; as functions of $x$.

Can I be sure that $f(x)$ increases monotonically as a function of $x$?

I can think of examples where both $n(x)$ and $d(x)$ increase monotonically, and $f(x)$ does not; but I cannot find such examples for $n(x)$ and $d(x)$ increasing and decreasing, respectively.

Answer 1

Asuming both $n(x)$ and $d(x)$ are never 0, you get for $x_1 > x_0$ by the respective $n(x_1) \ge n(x_0)$ and $d(x_1) \le d(x_0)$: $$ \frac{n(x_1)}{d(x_1)} \ge \frac{n(x_0)}{d(x_1)} \ge \frac{n(x_0)}{d(x_0)} $$

Answer 2

If $n(x),d(x)>0$ for all $x\in \mathbb R$, and $n(x)$ is increasing and $d(x)$ is decreasing, then $\frac{n(x)}{d(x)}$ is increasing.

The ratio may not be increasing otherwise. For example, consider $n(x)=-1$ for all $x\in\mathbb R$ and $d(x)=-1$ for $x<0$ and $d(x)=-2$ for $x\ge 0$.