Box "indeterminate power"

What do we do when we try to evaluate a limit and we end up with one of indeterminate powers 1^∞, 0⁰, or ∞⁰?

The standard procedure is to change this power into a product using the "e to logarithm" trick. This trick is based on the fact that for every positive real number x we have x = exp(ln(x)) and on the fact that logarithm changes powers into products. Namely, we transform the power as follows:

A^B = e^ln(AB ) = e^B⋅ln(A ).

Since this is just an algebraic transformation, the limit of the expression on the left is the same as the limit of the expression on the right. That one has the form of "e to something". Since exponential is a "nice function", we can "pull it out of the limit" and find the limit of the exponent first. This expression is a product, but what kind? If we start from an indeterminate power, we always get an indeterminate product:

We actually cheated a bit in the last case, see the note below.

Anyway, we transformed the problem of finding a limit of an indeterminate power into a problem of finding a limit of an indeterminate product. There we can apply the standard procedure (change the product into a ratio, then most likely use l'Hospital, see the box "indeterminate product"). After we find the limit of the exponent, we have to remember to put it back into the exponential.

Example:

Now we need to find the limit of the product x⋅ln(x) at 0 from the right. By a remarkable coincidence, we had exactly this example in the box "indeterminate product") and learned that it goes to zero. Therefore we can substitute back into the exponential.

Note that this transformation can be actually used anytime. For instance, if you do not remember (or trust) that in the limit algebra (1/2)^∞ = 0, you can do

(1/2)^∞ = e^∞ln(1/2) = e^−∞ = 0.

(We used the fact that ln(1/2) < 0.)

Note: An inquisitive reader should have gotten suspicious when we were transforming the power 0⁰ using the "e to ln" trick. Why? Because ln(0) does not exist, we cannot plug zero into the logarithm. What is happening here? The secret is that we cheated a bit when writing the power. General power requires that the base be positive, so in fact we do not have 0⁰, but (0⁺)⁰. Consequently, in the subsequent calculations we get ln(0⁺) and everything works as written above.

Why didn't we write the proper expression (0⁺)⁰ in the list of indeterminate powers? Because nobody bothers, it looks ugly, so we did not bother either. As a matter of fact, most calculus textbooks do not even bother to caution you about this one-sided stuff, we at least warned you. The good news is that you should not get bad results anyway. You should always check on 0 when putting it into logarithm in the limit algebra, and if the zero in the power were not of the positive type, we would find out anyway when working with logarithm and realize that there is something fishy.

In Solved Problems - Limits, these methods are used in this problem and this problem.

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