Convergence of an improper integral with generic continuous function

Question

I am trying to study the convergence or divergence of the following integral: $\int_{0}^{\infty} e^x b(x)dx$ where $b(x)$ is a continuous function that verifies $\lim_{x \to \infty}(b(x))=\alpha$ with $\alpha \in [-\infty,\infty]$. I have seen that, when $\alpha \neq 0$,it diverges by applying inequalities to the limit $\lim_{x \to \infty}(\int_{0}^{x} e^x b(x)dx)$ and showing its result is $\infty$ or $-\infty$ but I haven't been able to conclude in the case $\alpha=0$. Thanks.