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
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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