Problem:
An arbitrary circle can intersect the graph of in
Answer Choices:
A. at most points
B. at most points
C. at most points
D. at most points
E. more than points
Solution:
By definition, the graph of is the set of points in the rectilinear coordinate plane for which . Thus the graph of is the usual periodic wave along the -axis. Consider a circle tangent to the axis at the origin. If the radius is very large, the circle stays close to the axis for a long time and thus intersects the sine curve many times. By making the radius arbitrarily large, the number of intersection points may be made arbitrarily large.