How do epsilon-delta proofs prove the limit?

[Shaybay92]

I would really appreciate if someone could please explain to me how the final result of the proof of a limit actually PROVES the limit? http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/PrecLimSol.html

I was particularly looking at Solution 2 on this website. I understand how |x-10| was factorized out etc. and therefore it can equal delta... but when it says that if delta = epsilon/3, |x-10| < epsilon/3 and therefore |f(x)-35| is less than epsilon... how is this true? I mean we don't even know what epsilon is?

how does this even prove that the limit is 35

 

[Studiot]

I understand how |x-10| was factorized out etc. and therefore it can equal delta

Does it?

Do you understand the epsilon-delta argument. It is a two part formal method much used in mathematical analysis. Would this be your difficulty?

 

[tiny-tim]

Hi Shaybay92!

(have a delta: δ and an epsilon: ε )

… I mean we don't even know what epsilon is?

ε is anything (> 0).

Continuity is proved if, for any ε, you can find a δ (which depends on that ε) which works.

 

[Shaybay92]

ok but how does this proof tell us that there is a delta for any epsilon? i don't really see how if delta = cepsilon (c is some constant) then that is a proof...

 

[tiny-tim]

Hi Shaybay92!

(what happened to that δ and ε i gave you? )

ok but how does this proof tell us that there is a delta for any epsilon? i don't really see how if delta = cepsilon (c is some constant) then that is a proof...

because, if δ is defined as cε, then for any ε, there is a δ !

 

[Shaybay92]

so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?

 

[tiny-tim]

so as long as x is less than a particular multiple of ε (cε) then |f(x)-L| < ε?

That's right.

(Warning: although δ is usually a multiple of ε, somtimes it's something more awkward, like ε² )

 

[Studiot]

If you are going to study analysis in mathematics beyond high school level a good appreciation of the epsilon delta argument is esential.

I recommend taking the time to get a good grasp of it, try this

http://bobobobo.wordpress.com/2008/01/20/how-to-do-epsilon-delta-proofs-1st-year-calculus/

There are plenty more websites on the subject.