Delta-Epsilon Definition of a Limit...1

Calculus
Section 1.7

Can you work out 25 in step by step fashion as a guide for me to do a few more on my own?

 

25.

A function f is said to have a limit L as x approaches c, denoted , if for every ε>0, there exists a δ>0 such that |x−c|<δ implies |f(x)−L|<ε.

Then, to prove that , we must show that for any ε>0 there exists δ>0 such that |x−0|<δ implies .


Proof:
Let ε>0 be arbitrary, and let δ=min{ε,1}.

Suppose |x−0|=|x|<δ. Note that as δ≤1, we have |x|<1, meaning . Furthermore, as δ≤ε, we have |x|<ε. With those facts:

Thus, for an arbitrary ε>0, we have found a δ>0 such that |x−0|<δ implies
, meaning

.

 

Thank you but I will need more time to review what you are saying here.

 

I will do my best to do all the even numbers here. If I am wrong or get stuck, I would like your help or guidance.