SOLUTION: Is "lim x ->a f(x) = L" the same as for every epsilon>0 there exists a delta >0 such that if 0<|x-a|<delta then |f(x)-L|<epsilon Please explain Thanks

Question 988782: Is "lim x ->a f(x) = L" the same as for every epsilon>0 there exists a delta >0 such that if 0<|x-a| Please explain
Thanks
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

if and only if for all real numbers

there exists a real number

such that if

then

This just means that I can make the difference between f(x) and L as small as I want by selecting a sufficiently small difference between x and a.

Lets say you show me the function

, and I say to you that the limit as

gets close to 1 of

is 5. You, being a properly skeptical mathematician, say "Prove it."

So I say,

And then I say, I have proven my case because for any positive real number you choose, no matter how small, I can make the difference between f(x) and 5 smaller than your chosen number by choosing a difference between x and a to be smaller than one-half of your number.

John

My calculator said it, I believe it, that settles it

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