Does the limit exist for a function that oscillates between 2 and -2?

[appplejack]

Homework Statement

Evaluate the following limit
lim x->0 f(x) = 2, x rational, -2, x irrational

Homework Equations

The Attempt at a Solution

I tried to draw the graph of this question but couldn't. Do you think I have a function that oscillates between 2 and -2 violently?
My best guess is that the limit doesn't exist but I want to have better explanation than mine.

 

[Matterwave]

That looks like a slightly modified Dirichlet function to me. Between any 2 rational numbers is there an irrational number? And between any irrational number is there a rational number?

Also, what is this limit to? Limit as x goes to what?

 

[appplejack]

my apology. x->0.
There's no irrational number between 2 rational numbers. Vice versa no?

 

[Matterwave]

er...

Does your last statement make sense?

Between 1 and 2 is there an irrational number?

 

[Mr.Miyagi]

When you suspect that a limit does not exist, you can make use of the sequential criterion for limits. The criterion loosely states a limit \lim_{x→c}f(x)=L exists if and only if for every sequence (x_n) that converges to c it follows that \lim_{n→∞}f(x_n)=L.

So if you want to prove that the limit does not exist you can do so by showing that there exist two sequences (x_n) and (y_n) that converge to c and satisfy \lim_{n→∞}f(x_n)\neq \lim_{n→∞}f(y_n).

Does this make sense to you? Can you construct two sequences that would work in your case?

 

[appplejack]

Sorry. Could you explain it easier?

 

[gb7nash]

For a limit to exist, every time we make a sequence approach 0, the corresponding function values must approach the same number.

For example:

If I choose a sequence of rational numbers: 1,1/2,1/4,...

We get: f(1),f(1/2),f(1/4),...

So we get 2,2,2,... and this approaches 2.

However, if we look at e,e/2,e/4,...

We get: f(e),f(e/2),f(e/4),...

So we get -2,-2,-2,... and we get -2.

Since these are two different values, the limit does not exist.