Limits and Two Circles

Calculus
Section 2.3

We conclude Section 2.3 with this interesting but not so easy question involving two circles and the idea of limits. How on earth is this done? What happens to big R as little r tends to 0 from the right side along the x-axis?

Thanks

 

A stationary circle C1 have equation (x-1)^2+y^2=1 => radius=1, center (h,k)=(1,0)

Another circle of radius r is centered at (0,r)

R is the point of intersection of the line PQ and x-axis. (on given picture point (x,0)), so, for certain x value limit is 0

if C2 shrinks, P is changing position (goes down) while Q remain in same place

then direction of PQ is changing so that green line PQ will move up on the right and down on the left

ones point P comes to origin, C2 will disappear (becomes point), green line PQ goes from origin through point Q, so point Q will is intersection, R will go up and will not cross neither x-axis nor y-axis

so, limit is infinity

 

I am fascinated at the way you dive right into the problem and somehow find a solution even though you never saw the question before.