If the limit of a ratio is a constant greater than zero, the limit of the inverse ratio is also a constant greater than zero?

Question

Is it true that $$\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = c_1 \implies \lim_{x \rightarrow \infty} \frac{g(x)}{f(x)} = c_2$$ where $0 < c_1,c_2 < \infty$?

If yes, how can we prove that?

Edit: $f(x)$ and $g(x)$ are non-decreasing.

Answer 1

Hint. One may use the fact that $x \mapsto \dfrac1x$ is continuous over $(0,\infty)$.