Limits

Find the maximum value of n to the power 1/n and prove that it is a maximum.

Find all the turning points of y=x^{1/x} for x>0 and decide whether each is a maximum or minimum. Give a sketch of the graph.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Sketch the members of the family of graphs given by y = a^3/(x^2+a^2) for a=1, 2 and 3.

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.