Divergent? $\lim_{x\rightarrow\infty} A\left(\int_{0}^{\frac{x}{B}} te^{-t}\ dt\right)$

Question

Is the function below: $$ \lim_{x\rightarrow\infty} A\left(\int_{0}^{\frac{x}{B}} te^{-t}\ dt\right) $$ where $A$ and $B$ are both constants, a divergent or convergent one? Intuition when I glance at the limit tells me that it diverges, but every time I look at the graph or try out a few ideas with similar functions in my head, I begin to think that it's convergent depending on the values of $A$ and $B$. Do the values of $A$ and $B$ cause the function to diverge/converge? Thanks in advance!

Answer 1

We can use integration by parts to get $$\int_0^{x/B} t e^{-t}\, dt = \left[ -t e^{-t}\right]_0^{x/B} - \int_0^{x/B} (-e^{-t})\, dt = \frac{-x}{B} e^{-x/B} - \left[ e^{-t}\right]_0^{x/B} = \frac{-x}{B} e^{-x/B} - e^{-x/B} + 1$$

Assuming that $B > 0$, the limit of this function as $x\to \infty$ is $1$. Thus your original integral converges to the value $A$.