# elipson-delta definition of a limit to justify the answer.

• Sep 22nd 2011, 09:26 PM
LAPOSH42
elipson-delta definition of a limit to justify the answer.
So Far I found the limit:

Lim x->0 1/(x+1)=1

then this is the scratchwork:
1/(x+1)-1<E when 0< |x-0|< delta

Not sure if I did this inequality right:

|1/(x+1)-1|<E <==>|x/(x+1)|<E <==> |x/(x+1)|< 0E??

Do I have to find a number M??
1/|x+1| less than or equal to M

Proof:
|1/(x+1)-1|=|x|/|x+1| then I don't know how to keep going, not sure if it's 0E is correct (Wondering)
• Sep 22nd 2011, 09:48 PM
Prove It
Quote:

Originally Posted by LAPOSH42
So Far I found the limit:

Lim x->0 1/(x+1)=1

then this is the scratchwork:
1/(x+1)-1<E when 0< |x-0|< delta

Not sure if I did this inequality right:

|1/(x+1)-1|<E <==>|x/(x+1)|<E <==> |x/(x+1)|< 0E??

Do I have to find a number M??
1/|x+1| less than or equal to M

Proof:
|1/(x+1)-1|=|x|/|x+1| then I don't know how to keep going, not sure if it's 0E is correct (Wondering)

If $\displaystyle \displaystyle 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$, then $\displaystyle \displaystyle \lim_{x \to c}f(x) = L$.

In this case, you want to show $\displaystyle \displaystyle \lim_{x \to 0}\frac{1}{x + 1} = 1$, so you need to show that $\displaystyle \displaystyle 0 < |x - 0| < \delta \implies \left|\frac{1}{x + 1} - 1\right| < \epsilon$.

Some scratch-work...

\displaystyle \displaystyle \begin{align*} \left|\frac{1}{x + 1} - 1 \right| &< \epsilon \\ \left|\frac{x}{x + 1}\right| &< \epsilon \\ \frac{|x|}{|x + 1|} &< \epsilon \\ |x| &< |x + 1|\epsilon \end{align*}

Now suppose we restrict $\displaystyle \displaystyle |x| < \frac{1}{2}$ (say), then $\displaystyle \displaystyle -\frac{1}{2} < x < \frac{1}{2} \implies \frac{1}{2} < x + 1 < \frac{3}{2} \implies \frac{1}{2} < |x + 1|$, therefore

\displaystyle \displaystyle \begin{align*} |x| &< |x + 1|\epsilon \\ |x| &< \frac{1}{2}\epsilon \end{align*}

So if we let $\displaystyle \displaystyle \delta = \min\left\{\frac{1}{2}, \frac{1}{2}\epsilon\right\}$ and reverse the process, you will have your proof.
• Sep 23rd 2011, 04:16 AM
Siron
If you want to have more examples of proves using $\displaystyle \epsilon-\delta$ definition then I recommand you to take a look here: