Does L'Hospital's Rule still apply if the numerator and denominator go to infinity in different directions (i.e. positive infinity over negative infinity)?
That is exacly the same as $\displaystyle -\lim_{x\to\infty}\frac{2x}{e^{-x}}$. In general, if one factor goes to "$\displaystyle -\infty$", you can factor out a "-1" and turn the limit into $\displaystyle +\infty$. In other words, it doesn't matter whether the limit is $\displaystyle -\infty$ or $\displaystyle +\infty$.