#1 Calculus, Infinity, and all that
Posted: Wed Sep 13, 2006 11:22 pm
Today in my Calc II class, the professor was talking about log and exponential functions, and near the end, she put some integrals up on the board for us to do, namely, integral 1/x dx from 1 to e, from -e to -1, and -1 to e. The first two are simple, and the third one is obviously improper, though we haven't actually learned about those in class yet. If you simply evaluate each of them, they all come out to be 1, and then she says the third one is meaningless because of the asymptope. All of this, of course, is rather elementary.
When I did the third problem, though, applying the limit as a approaches 0 so we don't quite deal with infinity, the final step of the solution is
Lim_a->0 ln|a| - ln|-1| + ln|e| - ln|a|
Couldn't you cancel the ln|a|'s, and be left with just 1? That's what you would do to limits in any other case.
I talked to her after class, and I asked about it using the graph, saying that the area from -1 to 0 and 0 to 1 were the same, regardless of whether they were infinite or not, since the function was symmetric, and thus you could subtract the infinities in this case. She tried to explain that you couldn't because not all infinities are the same, which is a trivial concept that I understood anyway. I said that in this case, they were, and she pretty much said that you just can't do it that way.
So I'm left unsatisfied. I understand that infinity is just a concept and not really a quantity, but...Why aren't the areas equal? -1 to 0 is the same distance as 0 to 1 and the function is symmetric about the origin, so the two regions should be equal in area. Or am I just going to have to settle for "It just doesn't work like that"?
When I did the third problem, though, applying the limit as a approaches 0 so we don't quite deal with infinity, the final step of the solution is
Lim_a->0 ln|a| - ln|-1| + ln|e| - ln|a|
Couldn't you cancel the ln|a|'s, and be left with just 1? That's what you would do to limits in any other case.
I talked to her after class, and I asked about it using the graph, saying that the area from -1 to 0 and 0 to 1 were the same, regardless of whether they were infinite or not, since the function was symmetric, and thus you could subtract the infinities in this case. She tried to explain that you couldn't because not all infinities are the same, which is a trivial concept that I understood anyway. I said that in this case, they were, and she pretty much said that you just can't do it that way.
So I'm left unsatisfied. I understand that infinity is just a concept and not really a quantity, but...Why aren't the areas equal? -1 to 0 is the same distance as 0 to 1 and the function is symmetric about the origin, so the two regions should be equal in area. Or am I just going to have to settle for "It just doesn't work like that"?