Suppose one has $$F(b)= \int_{0}^{\infty} f(x, b)e^{g(x, b)}dx. $$ Next, suppose that $F(0) = \infty$ and $g(x, 0) = 1.$
Further suppose that $ \lim_{b \rightarrow 0}bF(b) $ converges.
Finally, let $$ \int_0^{\infty}\lim_{b\rightarrow 0}(bf(x, b)e^{g(x, b)})dx = \infty. $$ Can one say that $$ \lim_{b \rightarrow 0}bF(b) = e\lim_{b \rightarrow 0}\int_{0}^{\infty} bf(x, b)dx?$$
It seems to me that one cannot swap the limit right away, but perhaps it could be true and proven with careful derivation? Please, prove or disprove it.