Indicators that a limit does/does not exist

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Discussion Overview

The discussion revolves around the evaluation of limits in calculus, specifically questioning the reliability of certain methods for determining whether a limit exists for various types of functions. Participants explore examples of functions that challenge traditional limit evaluation techniques, including the Dirichlet function and step functions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant outlines a method for evaluating limits based on the behavior of the function at a point, noting specific cases where limits may or may not exist.
  • Another participant introduces the Dirichlet function as a counterexample, suggesting that the limit does not exist as a approaches 3.
  • A later reply agrees with the identification of the Dirichlet function and reiterates that the limit does not exist, while also presenting a simpler counterexample using a step function.
  • It is suggested that the original method for evaluating limits assumes continuity, which may not hold for all functions, thus questioning its general applicability.
  • One participant argues that the proposed indicators for limit evaluation are only valid in cases involving continuous functions and fractions.

Areas of Agreement / Disagreement

Participants express disagreement regarding the applicability of the limit evaluation method for all functions, with some asserting that it fails for certain discontinuous functions while others challenge the initial assumptions about continuity.

Contextual Notes

The discussion highlights limitations in the traditional methods of limit evaluation, particularly concerning functions that exhibit discontinuities or are defined piecewise. The assumptions underlying the method's applicability are not universally valid.

newageanubis
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Hey everyone,

When I first started learning calculus, I was taught that the first thing to do when asked to evaluate a limit as x -> a of f(x) is to evaluate f(a). If f(a) is of the form 0/0, then no conclusion can be made about the limit, and the expression needs to be manipulated by factoring, rationalizing, etc. before a conclusion can be made. If f(a) i of the form k/0, where a is a real number and k≠0, then the limit doesn't exist. Finally, if f(a) = k, where k is a real number, then the limit exists and is equal to k. But high school calculus only dealt with functions that behave nicely: polynomials, rational functions, trig functions, exponential/log.

My question is this: does this "method" of discerning the nature of a limit work for all f(x)?
 
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Consider the following function: Let f(a) = 1 when a is rational, and let f(a) = 0 when a is irrational. What is the limit as a approaches 3?
 
I believe that's the Dirichlet function, and I don't think that limit exists...:S
 
newageanubis said:
I believe that's the Dirichlet function, and I don't think that limit exists...:S

Yes on both accounts.

A simpler counterexample would be the step function: f(x) = 0 when x < 0, f(x) = 0.5 when x = 0, and f(x) = 1 when x > 0. The problem with the method lies in
the first thing to do when asked to evaluate a limit as x -> a of f(x) is to evaluate f(a)
f(a) can always be made to exist, by simply defining f(a)=0, without affecting the limit. In other words, you are making the assumption that f(x) is continuous to begin with, so you test f(a).
 
Last edited:
In other words, your "indicators" work only for the case where you have a fraction, with one continuous function over another.
 

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