How Do I Determine the Epsilon for a Limit in Analysis?

[semidevil]

now, this is not a calculus course...this is an analysis course, so I don't know if I'm suppose use the calculus techniques...

but anyways,

to show that if a limit exists, I need to show that for all e > 0, there exists K such that n >= k, then |x(n) - x| < e.

and usually, to verify that there is a limit, I would need to know what e is.

so how do I find e?

i.e, n / (n+1)

 

[cogito²]

sometimes limits can be difficult to determine but it completely depends on the situation. you don't always need to know what a limit is to determine if it exists...but that depends on cauchy sequences and completeness.

to figure out the limit one way is to just look at the numbers in the sequence. you could even use a calculator for this--one of the few times i even consider using it. there's a lot of ways for a lot of different sequences. some are easy to see and some are very hard to determine. it really varies from sequence to sequence.

 

[matt grime]

You don'd find e, you must show that given any e you can then... etc

n/(n+1) obviously tends to 1, since it equals 1+1/n, hence given any e, we must show that there is a K, such that for all n>K, |1+1/n - 1| <e, ie 1/n <e

let K=ceiling(1/e), then for n > K >= 1/e, 1/n< 1/K < e

done.

you don't get to choose e, you get to choose K dependent on e.

 

[HallsofIvy]

Did you notice that YOU said "for all e"? You don't FIND e- you have to show how you would find k for ANY GIVEN e.

 

[semidevil]

ok, maybe I worded it wrong...I guess I wanted to show you guys that i"m not a slacker and I did think about the problem...keke.

but anyways, I guess a more appropriate quesetion is...when they askk you to find the limit, how do you do it?

like, to show that a limit exists, there is a definition, and you need to show it. To find the actual limit, what is the thought process?

ok, I think that makes more sense.

 

[arildno]

There exist NO foolproof technique to find the number which might be the limit.
What you have at your disposal, is a technique to determine whether a chosen number is the limit or not.

 

[matt grime]

I might slightly dispute that: there is no technique I know per se that proves whether a chosen number is the limit, or whether one exists. There are several things to try, but no one that is guaranteed to prove fruitful apart from hard work.

There are, obviously, several useful results to learn about general limits such as products, sums and quotients behave as you want.

 

[arildno]

OK, I agree; "technique" was a very poor word choice.
We need to prove whether a given number satisfy the definition of a limit; we do not have at our disposal an all-purpose technique which might help us in this.

 

[semidevil]

ok, so for example, x(n) := n/(n+1).

how do you find the limit?

the answer is 1...but how do you work through it? what's the thought process?

 

[arildno]

You must check if "1" satisfies the properties that a limit must have.
That's the thought process involved.

 

[semidevil]

You must check if "1" satisfies the properties that a limit must have.
That's the thought process involved.

no no, I'm really sorry to be a pain...


what I meant is...given the problem...I know the limit is 1. But what if I didn't know that the limit is 1.

how do I figure it out?

 

[Gokul43201]

$$\lim_{n->\infty} \frac {n}{n+1} = \lim_{n->\infty} \frac{1}{1 + {\frac{1}{n}}}$$

Can you take it from there ?

 

[arildno]

But what if I didn't know that the limit is 1.

how do I figure it out?

You wouldn't be able to, in the general case.
You might be able to prove that a limit has to exist, though.