Does the Limit Exist for a Piecewise Function at x=1/2?

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SUMMARY

The limit of the piecewise function defined as f(x) = 2x for rational x and f(x) = 4 - 2x for irrational x does not exist as x approaches 1/2. As x approaches 1/2, the function oscillates between values close to 1 and 3, confirming that the limit cannot converge to a single value. The reasoning is supported by the fact that rational inputs yield outputs near 1, while irrational inputs yield outputs near 3, demonstrating the absence of a limit at this point.

PREREQUISITES
  • Understanding of piecewise functions
  • Knowledge of limits in calculus
  • Familiarity with rational and irrational numbers
  • Ability to apply the epsilon-delta definition of limits
NEXT STEPS
  • Study the epsilon-delta definition of limits in detail
  • Explore the behavior of piecewise functions at discontinuities
  • Learn about the concept of oscillating limits in calculus
  • Investigate the implications of rational versus irrational inputs in function analysis
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Students studying calculus, particularly those focusing on limits and piecewise functions, as well as educators seeking to explain the concept of limits in a rigorous mathematical context.

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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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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.
 

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