Indicators that a limit does/does not exist

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In summary, the method of evaluating a limit by first evaluating f(a) and then manipulating the expression may work for nicely-behaved functions, but it is not reliable for all functions. Counterexamples such as the Dirichlet function and step function show that this method can lead to incorrect conclusions.
  • #1
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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  • #2
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?
 
  • #3
I believe that's the Dirichlet function, and I don't think that limit exists...:S
 
  • #4
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).
 
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  • #5
In other words, your "indicators" work only for the case where you have a fraction, with one continuous function over another.
 

1. What is an indicator that a limit does not exist?

An indicator that a limit does not exist is when the function's values approach different values from the left and right sides of the designated limit point. This is known as a "discontinuity" and means that the function does not have a single value at the limit point.

2. How can I determine if a limit exists at a particular point?

To determine if a limit exists at a particular point, you can use the limit definition: if the left and right hand limits exist and are equal, then the overall limit exists. Alternatively, you can graph the function to see if there are any discontinuities or holes at the designated limit point.

3. Can a limit exist if the function is not defined at the designated limit point?

Yes, a limit can still exist even if the function is not defined at the designated limit point. This is because the limit only considers the behavior of the function as it approaches the designated point, not the actual value at that point.

4. What does it mean if both the left and right hand limits do not exist?

If both the left and right hand limits do not exist, it means that the function experiences a "jump" or "break" at the designated limit point. This is known as an "essential discontinuity" and indicates that the limit does not exist.

5. Can a limit exist if the function approaches infinity?

Yes, a limit can still exist if the function approaches infinity as long as the function's value does not approach infinity at the designated limit point. In this case, the limit would be considered to be infinity or negative infinity.

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