Exercise 1. Using
Definition 5.1, explain why .
See text and/or instructor's solution manual.
Solution. Recall that is
compact notation for , and
that , and
for all .
Then, by definition , and so that
We are done.
Aside. The reason
we split off the constant term is
to avoid the unpleasant computation .
What is the mathematical definition of " ? In calculus we call it an indeterminate form."
For the computer software Mathematica it results in the following error message:
For the computer software
it results in the following calculation:
In Section 5.2 we will see that log(z) is defined, for . In Section 5.3 we will define , for .
If you are curious about the "
controversy you can read the paragraph Zero
to the zero power" on Wikipedia.org
the URL is: http://en.wikipedia.org/wiki/Exponentiation#Powers_of_zero.
Another paragraph Zero to the zero power is found on the page http://en.wikipedia.org/wiki/Defined_and_undefined.
This solution is complements of the authors.