I have tried understanding this proof at least 3 times now thinking it would clear up eventually. It has not.
I don’t know why we have to assign these special variables epsilon and delta in the first place, why not a and b, or any other regular letter? is there some history with epsilon/delta?
Can someone explain it’s significance in it’s most basic form? like before you even try applying it to a derivative using the fundamental limit theorem?
In: Mathematics
First, Greek letters are fairly commonly used to represent mathematical variables, constants, and other concepts. In this context, episilon was historically used to represent infinitessimals and delta was historically used to represent change.
Delta is a common term for difference. So it is used several places in mathematics, for example calculus, to denote a very small difference between two inputs to a function. Epsilon is the next letter in the Greek alphabet from delta. So for the same reason you would use a and b, i and j, x and y, etc. it is natural to use delta and epsylon together.
the names are just convention. to make it easier to understand/identify what they mean.
like you could call your smartphone a “handheld computer with a touchscreen, wifi and telephone capabilities, a built-in harddrive…..” or you could say it’s “a smartphone” and most people would know. (in a typical math proof it is however usually stated near the beginning that you are using such a device and from there onwards will just call it “smartphone”).
they’re not used as variables. they’re a definition of what a limit is.
essentially what the epsilon delta criterion says is
“if you pick a value X and wiggle about that value by about delta-much – or ANYTHING smaller than delta, then the function wont behave wildly but instead will just also wiggle a tiny amount of epsilon or less”
this means that if you pick any epsilon (usually this is an assumed tiny tiny number) then you can find a value for delta where all variations of your main variable by delta or less wont change the function by more than epsilon.
probably a counterexample is helpful
look at f(x) = 1/x around x=0
if you go to -0.00001 you get a wildly different result than if you go to +0.00001 and the closer you move from both sides to 0 the further apart your results will be.
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