Proving a limit exists

**JasonW** · February 25th 2008, 05:30 PM

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!

**ThePerfectHacker** · February 25th 2008, 05:55 PM

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.

**JasonW** · February 25th 2008, 06:01 PM

I feel like I should know this already, but what does "Q" represent?

**ThePerfectHacker** · February 25th 2008, 06:02 PM

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.

**JasonW** · February 25th 2008, 06:06 PM

Got it, thanks!

Thread: Proving a limit exists

LinkBack

Thread Tools

Display

Proving a limit exists

Similar Math Help Forum Discussions

proving if such transformation exists

[SOLVED] limit of function exists, but limit of its derivative doesn't.

Proving limit exists

Proving a maximum exists

Proving a supremum exists.

Search Tags