area being finite in $\int_{0}^\infty \alpha\frac{x}{\beta+x}dx<\infty.$

Question

For $x>0$, for what values of $\alpha$ and $\beta$, do we have:

$$\int_{0}^\infty \alpha\frac{x}{\beta+x}dx<\infty.$$

This is known as the saturaion-growth model specification in nonlinear regression.

I would like to know for what values the area under the curve is finite or infinite when $x$ takes strictly positive values.

How do we obtain the threshold for this? What is the best way to go about this?

Answer 1

$\alpha \frac x {\beta +x} \to \alpha $ as $x \to \infty$. So the integral can be finite only when $\alpha =0$.

Answer 2

$$I_p=\int_{0}^p\frac{\alpha x}{\beta+x}dx=\alpha (\beta \log (\beta )-\beta \log (\beta +p)+p)$$ if $\Re(\beta )>0\lor p+\Re(\beta )\leq 0\lor \beta \notin \mathbb{R}$.

If these conditions are fulfilled, then, by Taylor $$I_p=\alpha p+\alpha \beta \log \left(\frac{\beta }{p}\right)-\frac{\alpha \beta ^2}{p}+O\left(\frac{1}{p^2}\right)$$ If you work in the real domain, then ...