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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

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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