Use of definitions to prove limits

• Feb 7th 2010, 09:16 AM
CrazyCat87
Use of definitions to prove limits
How would you prove the following limit using the epsilon-delta definition of a limit,

$\lim_{x\to1} \frac{x}{1+x} = \frac{1}{2}$
• Feb 7th 2010, 09:25 AM
VonNemo19
Quote:

Originally Posted by CrazyCat87
How would you prove the following limit using the epsilon-delta definition of a limit,

$\lim_{x\to1} \frac{x}{1+x} = \frac{1}{2}$

Where are you stuck?
• Feb 7th 2010, 06:30 PM
CrazyCat87
Quote:

Originally Posted by VonNemo19
Where are you stuck?

Sorry, here is what I have so far
$|\frac{x}{1+x} - \frac{1}{2}| = |\frac{2x-x-1}{2x+2}| = |\frac{x-1}{2x+2}| = |\frac{1}{2x+2}||x-1|$

And we know $|x-1| \leq \delta$ .....

That's all I have right now. I can't figure out where to go from there
• Feb 7th 2010, 06:41 PM
Chris L T521
Quote:

Originally Posted by CrazyCat87
Sorry, here is what I have so far
$|\frac{x}{1+x} - \frac{1}{2}| = |\frac{2x-x-1}{2x+2}| = |\frac{x-1}{2x+2}| = |\frac{1}{2x+2}||x-1|$

And we know $|x-1| \leq \delta$ .....

That's all I have right now. I can't figure out where to go from there

Fix $\delta =1$ so we can find an upper bound for $\frac{1}{\left|x+1\right|}$:

$\left|x-1\right|<1\implies -1.

Therefore $\frac{1}{\left|2x+2\right|}\left|x-1\right|=\frac{1}{2\left|x+1\right|}\left|x-1\right|<\frac{1}{2}\left|x-1\right|<\varepsilon$. Therefore, we pick $\delta=\min\{2\varepsilon,1\}$

Does this make sense?
• Feb 7th 2010, 07:18 PM
CrazyCat87
Quote:

Originally Posted by Chris L T521
Fix $\delta =1$ so we can find an upper bound for $\frac{1}{\left|x+1\right|}$:

$\left|x-1\right|<1\implies -1.

Therefore $\frac{1}{\left|2x+2\right|}\left|x-1\right|=\frac{1}{2\left|x+1\right|}\left|x-1\right|<\frac{1}{2}\left|x-1\right|<\varepsilon$. Therefore, we pick $\delta=\min\{2\varepsilon,1\}$

Does this make sense?

Ok yea the only thing that never makes sense to me is this step
$\left|x-1\right|<1\implies -1

how do you go from $-1 to $1
• Feb 7th 2010, 07:21 PM
Chris L T521
Quote:

Originally Posted by CrazyCat87
Ok yea the only thing that never makes sense to me is this step
$\left|x-1\right|<1\implies -1

how do you go from $-1 to $1

Whoops...that should be $1, but even then, the upper bound on $\frac{1}{\left|x+1\right|}$ remains the same.