Simple Question: division of sums

Question

I am a little bit confused about the following simple task. Given some functions $f(x), g(x), h(x), l(x), m(x)$.

We know that $\frac{f(x)}{g(x)}= m(x)$.

We further know that $h(x), l(x)$ are asymptotically smaller than $f(x), g(x)$ and $g(x)$ is asymptotically smaller than $f(x)$.

Now I would like to show that

$$\frac{f(x)+h(x)}{g(x)+l(x)}= m(x) + o(f(x)).$$

How does one show this? It should be true, shouldn't it?

Answer 1

Note that $\frac{f+h}{g+l} - \frac{f}{g} = \frac{h}{g+l} - \frac{fl}{g(g+l)} \in o(1) + o(f) = o(f)$. Thus $\frac{f+h}{g+l} = \frac{f}{g}+o(f) = m + o(f)$.