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

Algebra ->  Equations -> 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      Log On


   

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) About Me  (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