Evaluate the following problem and determine if it converges or diverges. If it converges, calculate what it converges to.

$\displaystyle \displaystyle\int^0_{-\infty} xe^x\,dx$

$\displaystyle \displaystyle\int^0_{-\infty} xe^x\,dx=\lim_{t\to-\infty}\int^0_t xe^x\,dx=\lim_{t\to-\infty}(xe^x-e^x)\bigg|_{0}^{t}$

$\displaystyle =[e^0(0-1)]-[e^{-\infty}(-\infty-1)]$

$\displaystyle =[-1]-[0(-\infty)]{}$

I know the answer is that it conveges to -1, but wouldn't it diverge since I end up with $\displaystyle -1-(0*\infty)$ ?

Thanks!