1. ## Limits

If f(x)=x/2 when x belongs to rational numbers
=0 when x belongs to irrational numbers
Can limit x reaches 0 exist?

And If g(x)=0 when x belongs to rational numbers
=x+1 when x belongs to irrational numbers
Can limit x reaches 0 exist?

2. ## Re: Limits

I'll help with the first - similar logic applies to the second. The definition of the limit requires that as x approaches the 0 the value of f(x) gets "closer and closer" to the limit 'L,' meaning that while it doesn't have to actually equal L it gets closer than any arbitrary error (epsilon) that you can name. For this problem you first need to know that for any value of x there are an infinite number of rational and irrational numbers between 0 and x, so the value of f(x) alternates between 0 and x/2 an infinite number of times. But you can see that as x gets close to 0 the value of x/2 also gets close to zero. So if I challenge you by asking what value of x will guarrantee that f(x) is less than, say, 0.01, you can see that this is true if x<= 0.02. For all values less than 0.02 the value of f(x) is less than 0.01. Thus this function passes the "epsilon-delta" test - for any epsilon you can name (0.01 in this exampe) it's possible to find a delta below which f(x)< epsilon.

Now think about the second problem - as x gets small and approaches zero what value - if any - does f(x) approach?

3. ## Re: Limits

Originally Posted by Kristen111111111111111111
If f(x)=x/2 when x belongs to rational numbers
=0 when x belongs to irrational numbers
Can limit x reaches 0 exist?
Yes, it should be clear that the limit is 0. Given any epsilon> 0, take delta to be a number less than 2epsilon. If |x- 0|= |x|< delta, then either x is irrational, so that f(x)= 0< epsilon or x is rational so that |f(x)|= |x|/2< delta/2< epsilon. In either case, |f(x)- 0|= |f(x)|< epsilon.

And If g(x)=0 when x belongs to rational numbers
=x+1 when x belongs to irrational numbers
Can limit x reaches 0 exist?
Suppose x is an irrational number, very close to 0. What can you say about f(x)? Suppose x is a rational number, very close to 0?

4. ## Re: Limits

Thank you so much. From your explanations I got that limit x reaches 0 exists for f(x) and limit x reaches 0 does not exist for g(x).
And just to clarify I need to know whether a limit exist for a function defined only on the set of rational numbers?
For example f(x) = x/2 , for all x belongs to rational numbers , does the limit x reaches 0 exist?

5. ## Re: Limits

Originally Posted by Kristen111111111111111111
Thank you so much. From your explanations I got that limit x reaches 0 exists for f(x) and limit x reaches 0 does not exist for g(x).
And just to clarify I need to know whether a limit exist for a function defined only on the set of rational numbers?
For example f(x) = x/2 , for all x belongs to rational numbers , does the limit x reaches 0 exist?
Depends how you define the value off(x) of x = irrational numbers. If you say f(x) is undefined for irrational numbers then no, there is no value for the limit.

6. ## Re: Limits

Yeah got that. Thank you