# Precise definition of a 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 dont even know what epsilon is?

how does this even prove that the limit is 35

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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?

Homework Helper
Hi Shaybay92! (have a delta: δ and an epsilon: ε )
… I mean we dont 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 dont really see how if delta = cepsilon (c is some constant) then that is a proof...

Homework Helper
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 dont 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?

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