# 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

.

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