# Thread: limit of a sequence

1. ## limit of a sequence

[img]http://img708.imageshack.us/img708/4042/analysis999.jpg[/img]
I stuck halfway through, any help would be appreciated.
-----------------------------------
Let ε>0 be given.
|x_n - (9/2)| = ......
simplified to:
39/[(4n^3) + 10]
Im stuck here, what do i do next?

2. Originally Posted by rlkmg
[img]http://img708.imageshack.us/img708/4042/analysis999.jpg[/img]
I stuck halfway through, any help would be appreciated.
-----------------------------------
Let ε>0 be given.
|x_n - (9/2)| = ......
simplified to:
39/[(4n^3) + 10]
Im stuck here, what do i do next?
Well you need to find $N$ in terms of $\epsilon$

Note that

$\displaystyle \bigg|\frac{39}{4n^3+10} \bigg|\le \frac{39}{4}\bigg|\frac{1}{n^3} \bigg|$

Now set this equal to epsilon to find big $N$

$\displastyle \frac{39}{4}\frac{1}{N^3} =\epsilon \iff N^3=\frac{39}{4\epsilon} \iff N=\sqrt[3]{\frac{39}{4\epsilon}}$

Now you can start the formal proof e.g

Let $\epsilon > 0$ and and let $N=\sqrt[3]{\frac{39}{4\epsilon}}$ then for $n > N$...

3. Originally Posted by TheEmptySet
$\displaystyle \bigg|\frac{39}{4n^3+10} \bigg|\le \frac{39}{4}\bigg|\frac{1}{n^3} \bigg|$
Thanks so much, this is the part that I dont understand, could you explain how you got from the left to the right of the above equation? I've seen your technique in many examples but i still dont understand how you did the above step. especially getting rid of the 10

Thanks for your help, appreciated

4. Originally Posted by rlkmg
Thanks so much, this is the part that I dont understand, could you explain how you got from the left to the right of the above equation? I've seen your technique in many examples but i still dont understand how you did the above step. especially getting rid of the 10

Thanks for your help, appreciated
okay I think you are over thinking this lets look at a few numerical examples.

The idea is if you make the denominator of a fraction smaller (divide by a smaller number) the fraction gets bigger for example

$\displaystyle \frac{1}{12}=\frac{1}{2+10} \le \frac{1}{2}$

So the idea is to make the fraction easier to work with by replacing it with a bigger but simpler fraction.

5. Thanks, i tried a few examples and i've got more of a hang of it