A Curious Geometric Limit
Consider the following diagram in which we have circle X of radius 3 centered at (3,0) and circle O of radius k centered at the origin. Call the intersection of the these circles in the first quadrant B. Let A be the intersection of circle O with the y-axis, and extend line AB until it intersects the x-axis at E. Our goal is to show that .
Consider the diagram below.
Let . Some simple angle chasing, using properties of isosceles triangles and right triangles, gives us that .
Since AO = k by definition, we have , and so OE = .
Since OB = k is the base of isosceles triangle OXB, we can drop an altitude from X to OB and find that . This gives us .
So OE = = . By the double angle formula for sine, we have
Making this substitution, and by writing tangent as the quotient of sine and cosine, we get
OE = .
Finally, since as , we have