# Math Help - Question Concerning L'Hospital's Rule

1. ## Question Concerning L'Hospital's Rule

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)?

2. Originally Posted by cusseta
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)?
What is the limit this is in regards to?

3. For example: The limit as x approaches negative infinity of ( 2x / -e^-x ).

In this example, the numerator approaches positive infinity, but the denominator approaches negative infinity.

4. Originally Posted by cusseta
For example: The limit as x approaches negative infinity of ( 2x / -e^-x ).

In this example, the numerator approaches positive infinity, but the denominator approaches negative infinity.
$\displaystyle\lim_{x\to\infty}-\frac{2x}{e^{-x}}=\lim_{x\to\infty}-\frac{2x}{\frac{1}{e^x}}\cdot\frac{e^x}{e^x}=\lim_ {x\to\infty}=-2xe^{x}=\cdots$

5. That is exacly the same as $-\lim_{x\to\infty}\frac{2x}{e^{-x}}$. In general, if one factor goes to " $-\infty$", you can factor out a "-1" and turn the limit into $+\infty$. In other words, it doesn't matter whether the limit is $-\infty$ or $+\infty$.

6. Thanks! I knew it was something ridiculously simple!