Math vs. the “Real World”

There is a difference between mathematics and the “real world”: In the real world you are allowed to do whatever you want, as long as there is no law against doing it; in mathematics, you cannot do something unless there is a law that says you may.

A question that comes up often on the AP Calculus Community bulletin board concerns the divergence of the improper integral \displaystyle \int_{{-1}}^{1}{{\frac{1}{x}dx}}.

There are several mistakes students make when computing this integral.

First, they may not realize this is an improper integral and compute incorrectly:

\displaystyle \int_{{-1}}^{1}{{\frac{1}{x}dx}}=\ln \left| 1 \right|-\ln \left| {-1} \right|=0-0=0

Since the laws concerning improper integrals do not allow this, you may not do it.

Or, they might think that it does converge to zero by the symmetry of the graph. There is no law (theorem) that permits calculating limits based on the appearance of a graph.

Finally, and most often, they may start out following the rules but go astray. The law says you must find deal with the discontinuity at x = 0 by using one-sided limits:

\displaystyle \int_{{-1}}^{1}{{\frac{1}{x}dx}}=\underset{{a\to 0-}}{\mathop{{\lim }}}\,\int_{{-1}}^{a}{{\frac{1}{x}dx}}+\underset{{b\to 0+}}{\mathop{{\lim }}}\,\int_{b}^{1}{{\frac{1}{x}dx}}

=\underset{{a\to 0-}}{\mathop{{\lim }}}\,\left( {\ln \left| a \right|-\ln \left| 1 \right|} \right)+\underset{{b\to 0+}}{\mathop{{\lim }}}\,\left( {\ln \left| 1 \right|-\ln \left| b \right|} \right)

=\left( {-\infty -0} \right)+\left( {0-(-\infty )} \right)=\infty -(\infty )=0

The mistake is subtler here. It is correct to say that  =\underset{{a\to 0-}}{\mathop{{\lim }}}\,\ln \left| a \right|=-\infty , but what that really means is that the limit does not exist (DNE). Then in the last line above you cannot say

(does not exist) – (does not exist) = 0.

You cannot subtract something that does not exist from something else that does not exist. As soon as you see that one of the limits does not exist, the entire limit does not exist. (That’s the law.) There is no algebra/calculus theorem that permits the addition of two divergent integrals, therefore, it is not correct to add them.

Students need help in understanding this. Here are three ways to think about it.

  1. Infinity is not a number. When you say the integrals equal infinity and negative infinity, you must stop. Just because it looks like something minus the same thing is zero, you cannot do this, because you’re not working with numbers. In fact, the integrals do not exist, so you cannot add them – there’s nothing to add.
  2. “Infinity” is a short, and correct, way of expressing the limits as you approach zero for this function from the right or left. But, you must remember that infinity is a shorthand for DNE, not for some really large number and its opposite.
  3. Infinity minus infinity (\infty -\infty ) is an indeterminate form. Some indeterminate forms of this type converge, if you can find some additional algebra/calculus to do on them (such as L’Hospital’s Rule in some cases). For this example, such algebra/calculus does not exist (no pun intended)

So, in conclusion \displaystyle \int_{{-1}}^{1}{{\frac{1}{x}dx}} does not converge!

This question was discussed recently on the AP Calculus bulletin board. The two items below were included and may help your students understand what’s going on with infinity. The are by Stu Schwartz.

Thank You Stu!


2 thoughts on “Math vs. the “Real World”

  1. Can you clarify something for me? You stated “It is correct to say that lim a->0- (ln|a|) = infinity” (sorry for the format). I’m wondering why the limit would not be negative infinity?
    Thanks so much for your regular blogs, they have been so helpful to me.


    • You are correct. I’ve changed the two lines involved. Also, \underset{{b\to 0+}}{\mathop{{\lim }}}\,\ln \left| b \right|=-\infty . While this is a correct use of infinity, remember that infinity is not a number; it means the expression can be made larger or smaller than any number. Thanks for catching that.


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