Problem:
Explore the limit

for various values of n and m.


Visualization:
Enter an integer value for n:

Enter an integer value for m:

Try various values for n and m and see what the limits appear to be. Make a conjecture that you think that will happen for all values of n and m. Test your conjecture by choosing other values for n and m.

There are several reasons why this limit is important.

  1. The value of this limit when m = n = 1 is used to calculate the derivative of the sine function in a later lesson.
  2. The problem of calculating the limit of a quotient when the limits of both the denominator and the numerator are zero will be discussed several times in the calculus course. This is the first example that is ususally discussed in a calculus course where the numerator is not a polynomial.
In case you are wondering, this time we are choosing x to be powers of 1/2. The number N in the table indicates the power.