A Curious Geometric Limit

Inspired by Alexander Bogomolny (@CutTheKnotMath) and his curious geometric limit, I offer this trigonometric approach.

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

.