Let $g$ be a differentiable, nonnegative and decreasing function on $(0,\infty)$. Then $\int_k^{\infty} \, g(x) \, dx \approx 0$

Question

I am doing some work in probability and mathematical statistics, and I'm frequently engaged with nonnegative and decreasing functions on $(0,\infty)$. Some of the calculations would be smoother if the following conjecture is true, which I can see being true intuitively. So I formulated a conjecture, but I'm not sure how to prove it. Another obstacle is that I would like to have a more rigorous meaning as to what $\approx 0$ means.

Could someone help me or give me a hint as to how I can prove the following or provide a counterexample? I haven't gotten far.

$$\textbf{Conjecture}$$ Let $g$ be a differentiable, nonnegative and monotonically decreasing function on $(0,\infty)$. Then there exists $k \in \operatorname{dom}(g)$ such that $$\int_k^{\infty} \, g(x) \, dx \approx 0$$

Answer 1

$g(x)=1/x$, for $\displaystyle\int_{k}^{\infty}\dfrac{1}{x}dx=\infty$, so that is not true.