Limits and Two Circles

Joined
Jun 27, 2021
Messages
5,386
Reaction score
422
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

Screenshot_20220508-183745_Samsung Notes.jpg
 
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
 
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.
 

Members online

No members online now.

Forum statistics

Threads
2,521
Messages
9,844
Members
697
Latest member
RicoCullen
Back
Top