# Question Concerning L'Hospital's Rule

• Jan 24th 2011, 05:17 PM
cusseta
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)?
• Jan 24th 2011, 05:27 PM
dwsmith
Quote:

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?
• Jan 24th 2011, 05:44 PM
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.
• Jan 24th 2011, 05:48 PM
dwsmith
Quote:

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.

$\displaystyle\lim_{x\to\infty}-\frac{2x}{e^{-x}}=\lim_{x\to\infty}-\frac{2x}{\frac{1}{e^x}}\cdot\frac{e^x}{e^x}=\lim_ {x\to\infty}=-2xe^{x}=\cdots$
• Jan 24th 2011, 05:54 PM
HallsofIvy
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 $+\infty$.
• Jan 25th 2011, 10:09 AM
cusseta
Thanks! I knew it was something ridiculously simple!