\(\displaystyle 0 < |x-c| < Î´\) => \(\displaystyle |f(x) - L| < Îµ\) , \(\displaystyle \forall x\) maybe except c.

First of all do I understand what the definitions says? My understanding is this:

"The

**closer**x is at c the more

**precisely**f(x) will approach L." In order to define this mathematically we need two things. An number Î´ to precisely specify the words "The closer x is at c" and another number Îµ which precisely describes the words "the more precisely f(x) will approach L". If Îµ is really small, the more precise the limit will be.

**Some exercises which I'm trying to solve:**

Exercise 1:

Exercise 1 (continues):

Exercise 2:

For example in exercise 1 he finds out that: \(\displaystyle |x-1| < Îµ/5\)

Then i can not see why he chooses Î´=Îµ/5 . Is it because both in-equations are similar?

\(\displaystyle |x-1| < Îµ/5\)

\(\displaystyle 0 < |x-1| < Î´\)

so it seems like Î´ = Îµ/5 if you compare them. Is there a more mathematically procedure to actually show that?

Can you explain the solution of ex1 better that the book?

Thank you.