Real Analysis limit problem

In summary, the function f:ℝ→ℝ is defined as f(x)= 2x if x is rational and f(x)=4-2x if x is irrational. The limit as x approaches 1/2 does not exist, as the function oscillates between values close to 1 and 3, depending on whether x is rational or irrational. This can be shown using the definition of a limit or other theorems.
  • #1
gottfried
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0

Homework Statement


f:ℝ→ℝ is defined as f(x)= 2x if x is rational and f(x)=4-2x if x is irrational.
Is it true that lim x→1/2=1?


2. The attempt at a solution
Intuitively it seems that as x gets ever closer to 1/2 from either side that the function will oscillate between numbers very close to 1 and 3 and therefore there the limit doesn't exist.
Firstly is my intuition right?
Secondly how does one show this using the definition of a limit or some other theorem?
I thought about using the fact that all (xn)[itex]\subseteq[/itex]ℝ such that (xn)→1/2 would have to imply that (f(xn))→1. Again i can't formally show that this can't be.
 
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  • #2
Yes, that is correct. If x is a rational number close to 1/2, then f(x) is close to 2(1/2)= 1 and if x is an irrational number close to 1/2, then f(x) is close to 4- 2(1/2)= 3. The function does NOT get close to anyone number for all real numbers close to 1/2 and so the limit does not exist.
 

1. What is a limit in Real Analysis?

A limit in Real Analysis refers to the value that a function approaches as its input approaches a specific value. It is an essential concept in calculus and is used to describe the behavior of a function near a particular point.

2. How do you solve a limit problem in Real Analysis?

To solve a limit problem in Real Analysis, you must evaluate the function at the given input value and see what value the function approaches as the input gets closer to that value. This can be done by plugging in values that are closer and closer to the given input value and observing the trend of the function's output.

3. What is the difference between a one-sided and two-sided limit in Real Analysis?

A one-sided limit in Real Analysis refers to the value that a function approaches from only one direction, either the left or the right. In contrast, a two-sided limit considers the behavior of a function from both directions as the input approaches the given value.

4. Can a limit exist even if the function is not defined at that point?

Yes, a limit can exist even if the function is not defined at that point. This is because a limit only considers the behavior of the function near a specific point and does not depend on the value of the function at that point. As long as the function approaches a particular value as the input approaches the given point, the limit exists.

5. What are the common techniques used to evaluate limits in Real Analysis?

Some common techniques used to evaluate limits in Real Analysis include direct substitution, factoring, rationalizing, and using the properties of limits such as the sum, difference, and product rules. Other methods such as L'Hôpital's rule and Taylor series can also be used in more complex cases.

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