Finding a Counterexample to a Wrong Statement about Limits

[vibha_ganji]

Homework Statement

The number L is the limit of ƒ(x) as x approaches c if, given any epsilon greater than 0, there exists a value of x for which lƒ(x) - Ll is less than epsilon.

Relevant Equations

lf(x)-Ll < epsilon

I’m complete stuck on this problem. I am not sure how to start to find a counterexample to this statement.

 

[docnet]

another way is:

1. let $f=x$
2. show that the limit of $f$ at $a$ is $b$, where $b\neq a$, by finding an $x$ such that $|f(x)-b|<\epsilon$
3. conclude that the statement is false

 

[PeroK]

Homework Statement:: The number L is the limit of ƒ(x) as x approaches c if, given any epsilon greater than 0, there exists a value of x for which lƒ(x) - Ll is less than epsilon.
Relevant Equations:: lf(x)-Ll < epsilon

I’m complete stuck on this problem. I am not sure how to start to find a counterexample to this statement.

Do you think it might be a valid definition of a limit? If not, why not? What's wrong with it?

You need to see what's wrong with that definition, and then you can find a counterexample.

 

[FactChecker]

This is a homework problem, so we can only give hints and redirection to the work that you show us.
For counterexamples, look for functions that "wiggle around widely and very rapidly" as it approaches a point. There will be a sequence of x values that approach that point and whose function values are the same, but it wiggles around so that another sequence of x values approaches the same point and all have a second function value.

 

[pasmith]

Does this 'definition' actually require $x$ to be anywhere near $c$?

Your relevant equations section should really have included the actual definition of a limit.