# Limit of Functions

1. Apr 22, 2012

### renolovexoxo

1. The problem statement, all variables and given/known data

Negate the definition of the limit of a function, and use it to prove that for the function
f : (0; 1) --> R where f(x) 1/x, lim x-->0 f(x) does not exist.

2. Relevant equations

The limit of f at a exists if there exists a real number L in R such that for every e>0 there exists d>0 such that for every x in the interval with 0<|x-a|<d then |f(x)-L|<e.

3. The attempt at a solution

The limit of f at a does not exist if for all real numbers L in R such that for every e>0 there exists d>0 such that there exists xin the interval with 00<|x-a|<d then |f(x)-L|>e

2. Apr 22, 2012

### SammyS

Staff Emeritus
The following is true even if L is the limit at a .

For every δ > 0 there needs to be an ε > 0 (this ε usually depends upon the δ) such that there is some x0 for which 0 < |x0-a| < δ and |f(x0)-L|> ε .