Indicators that a limit does/does not exist

newageanubis

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)?

Number Nine

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?

newageanubis

I believe that's the Dirichlet function, and I don't think that limit exists...:S

pwsnafu

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
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).

HallsofIvy

In other words, your "indicators" work only for the case where you have a fraction, with one continuous function over another.