I have here below a mathematical statement, and it is False:
If $f(x)$ has a horizontal asymptote at $y = 0$ and $f(x)$ is always negative, continuous and increasing on $[1, \infty)$, then the area between the graph of $f(x)$ and the x-axis on $[3, \infty)$ must be finite.
I can't figure out an example of a function that behaves this way and the area works out to be infinite.
I find this very counter-intuitive. Hope some can figure out an example of a function of this type.
Regards,
As first noted in the comments, you can have for example
$$f(x)=-\frac{1}{x^p} \quad | \quad 0 < p \le 1$$
You can experiment with any given function $g(x)$ with a horizontal asymptote at say $y=b$. Just make either $g(x)-b \ \lor \ -g(x)-b$ and see if it meets the requirement of having infinite area
A non rational example would be
$$f(x)=-\frac{1}{\ln x}$$