# Proving a limit exists

• Feb 25th 2008, 05:30 PM
JasonW
Proving a limit exists
Need a little help on the homework in the calculus book...

I need to use an example to prove that Lim [F(x)+G(x)] as X approaches A may exist, even if neither Lim F(x) as X approaches A nor Lim G(x) as X approaches A exist.

I know this has something to do with the "Pinch" Theorem, but I really don't know where to go with this.

Thanks!
• Feb 25th 2008, 05:55 PM
ThePerfectHacker
Quote:

Originally Posted by JasonW
Need a little help on the homework in the calculus book...

I need to use an example to prove that Lim [F(x)+G(x)] as X approaches A may exist, even if neither Lim F(x) as X approaches A nor Lim G(x) as X approaches A exist.

Let $f(x) = \left\{ \begin{array}{c}1 \mbox{ if }x\in \mathbb{Q} \\ 0 \mbox{ if }x\not \in \mathbb{Q} \end{array} \right.$ and $g(x) = \left\{ \begin{array}{c}0\mbox{ if }x\in \mathbb{Q} \\ 1 \mbox{ if }x\not \in \mathbb{Q} \end{array} \right.$

Then for any point $a\in \mathbb{R}$ we have that $\lim_{x\to a}f(x)$ and $\lim_{x\to a}g(x)$ fail to exist while $\lim_{x\to a}f(x)+g(x)$ exists.
• Feb 25th 2008, 06:01 PM
JasonW
I feel like I should know this already, but what does "Q" represent?
• Feb 25th 2008, 06:02 PM
ThePerfectHacker
Quote:

Originally Posted by JasonW
I feel like I should know this already, but what does "Q" represent?

Rational numbers. Not "not Q" means an irrational number.
• Feb 25th 2008, 06:06 PM
JasonW
Got it, thanks!