Proving Non-Existence of a Limit: Solving 1/(x^2+x^3)

[lizielou09]

Homework Statement

does not exist.

Prove that the limit as x approaches 0 of 1/(x²+x³)

Homework Equations

The Attempt at a Solution

I know that I have to prove that the absolute value of 1/(x²+x³) - L is greater than or equal to epsilon for some delta. What do I do next?

 

[CompuChip]

The definition of limit says:
$$\exists L, \forall \epsilon > 0, \exists \delta > 0, \forall x \text{ s.t.} |x| < \delta: |f(x) - L| < \epsilon$$
where L is the supposed limit.

What is the negation of this?

 

[lizielou09]

The negation would be that the absolute value of f(x)-L is greater than or equal to epsilon. But how do I prove that there exists a delta for which that is true?

 

[JG89]

Why don't you just show that for all positive real numbers M, |1/(x^2 + x^3)| > M if x is taken sufficiently small?

 

[CompuChip]

Yes, but you should take care with the quantifiers: the negation of
$$\exists L, \forall \epsilon > 0, \exists \delta > 0$$
is
$$\forall L, \exists \epsilon > 0, \forall \delta > 0$$

I was stressing this because I think it is important that you do not fall into such logical traps.

Of course, if you just want to solve the question, follow JG's advice :)