So as the base gets closer to , the answer remains . So, from this perspective, it looks like ought to be equal to .

In conclusion: looking at it one way, should be defined to be . From another perspective, should be defined to be .

Of course, we can’t define a number to be two different things! So we’ll just say that is undefined — just like dividing by is undefined — rather than pretend that switches between two different values.

Here’s a more technical explanation about why is an indeterminate form, using calculus.

Part 1. As before,

.

The first equality is true because, inside of the limit, is permitted to get close to but cannot actually equal , and there’s no ambiguity about if . (Naturally, is undefined if .)

The second equality is true because the limit of a constant is the constant.

Part 2. As before,

.

Once again, the first equality is true because, inside of the limit, is permitted to get close to but cannot actually equal , and there’s no ambiguity about if .

As before, the answers from Parts 1 and 2 are different. But wait, there’s more…

Part 3. Here’s another way that can be considered, just to give us a headache. Let’s evaluate

Clearly, the base tends to as . Also, as , so that as . In other words, this limit has the indeterminate form .

To evaluate this limit, let’s take a logarithm under the limit:

Therefore, without the extra logarithm,

Part 4. It gets even better. Let be any positive real number. By the same logic as above,

So, for any , we can find a function of the indeterminate form so that .

In other words, we could justify defining to be any nonnegative number. Clearly, it’s better instead to simply say that is undefined.

P.S. I don’t know if it’s possible to have an indeterminate form of where the answer is either negative or infinite. I tend to doubt it, but I’m not sure.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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