Problem:
Explore numerically the following limits


Visualization:
Each of these limits gives the incorrect answer when we use the techniques developed so far.

Example 1.

Note, from the table, that when x = 0.000001, the value in the next column for the quotient is 0. This is not true.

The problem is that when a real number is used in the program, only 16 digits of the number are stored. So, when x = 0.000001 is stored in x3 + 8, the value which should be 8.000000000000000001 is stored as 8.000000000000000. Consequently, the expression becomes 0 instead of the correct value which is close to 1/12.

Most graphing calculators store numbers correct up to 8 or 9 digits only.


Example 2.

If you look at only the first ten values in the table for this limit then you would probably assert that the limit is 0. However, looking at the eleventh to the fifteenth values, the answer appears to be 7.0e-10 (in scientific notation).

It is easy to modify this example, to come up with an example that would need several hundred values to detect what appears to be the correct answer. What this example shows is that we cannot completely rely on numerical techniques to evaluate limits.


Example 3.

Since most of the values in the second coumn of the table are very small (remember that most of these are expressed using scientific notation). You might expect that the limit is 0. In fact, if you evaluate these numbers exactly then the numbers are almost all 0. The difficulty is that the numbers chosen for x are powers of 1/2. Look at what happens, if we choose numbers for x that are powers of 1/10.

From this table, it appears that the limit does not exist which, in fact, is the case.


We will develop techniques to come up with the exact answers for each of these limits.