Divergent Improper Integral

I could use some help with an integral. The problem asks to determine whether a given integral converges or diverges. I know that the answer is "diverges" but I'm not sure why.

$\displaystyle \int_{-\infty}^0 xe^{-2x} dx$

I used integration by parts to get: $\displaystyle [xe^{-2x} - \int e^{-2x} dx] = xe^{-2x}-e^{-2x}$

Then I evaluated $\displaystyle \lim_{b\to-\infty}[xe^{-2x}-e^{-2x}]_{b}^{0} = (-1)-(be^{-2b}-e^{-2b})$

I understood $\displaystyle \lim_{b\to-\infty}be^{-2b}$ to require the use of L' Hopital's rule which I applied as follows: $\displaystyle \lim_{b\to-\infty}\frac{b}{e^{2b}}=\frac{1}{2e^2b}=\frac{1}{-\infty}=0$ If this were true, the improper integral would evaluate to -1 - 0 = -1 and would converge. Can anyone tell me where I went wrong?