Exercise 1.  Using Definition 5.1, explain why  .  

Solution 1.

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:

[>  0^0;

                           1

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.