Intersecting Circles and Limits

This image shows two circles and a ray. The black circle, , is centered on the point (1, 0) and has a radius of one. The red circle, , is centered on the origin and has a decreasing radius, . Point is located at the top of circle , and point is the upper intersection point of circles and . Ray intersects the -axis at the point (, 0).
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Making a Prediction

What will happen to the point (, 0) as approaches zero? Explain your reasoning.

Coming up with a Plan

Now that we've made a prediction, let's see if we can prove what will happen to the point. Come up with a plan. How can we prove what will happen to the point (, 0) as the radius decreases to zero?

Part 1: Identifying Preliminary Information

To determine what happens to the point (, 0), we need to find the equation of line . In order to do this, we must identify the coordinates of points and . What are the coordinates of point ?

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Circle 1

In order to find the coordinates of point we must identify the equation of each circle. Which equation describes circle ?

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Circle 2

Which equation describes circle ?

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Part 2: Using a System of Equations

Now that we know the equations that describe the two circles we can find their intersection point. To do so, we must solve both equations simultaneously. Start by identifying the x-coordinate of the intersection point.

Finding the x-coordinate

What is the x-coordinate of the intersection point? Use the hints below if you get stuck.

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Hints

Finding the y-coordinate

Now that we have the x-coordinate of the intersection point, we can find the y-coordinate. Substitute the x-value back into either equation and solve for y. What is the y-coordinate of the intersection point? Use the hints below if you get stuck.

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Hints

The Coordinates of Point Q

We have now solved the system of equations and identified the coordinates of point Q. It is important to note that there are many equivalent expressions that also represent the coordinates of point Q. For our purposes, however, this will be the form that we use. Recall that we are trying to find an equation of the line associated with ray PQ. The next step in this process is to find the the slope of the line.

Part 3: Using the Equation of PQ

Since we have the coordinates of points P and Q, we can identify the slope of the line. The slope between any two points can be found using the formula           Which expression represents the slope between the two points? Use the hints below if you get stuck.

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Hints

Writing and Solving an Equation

Now that we have identified the slope of line PQ, we can write an equation that describes it. Since we are interested in identifying the value of , substitute y = 0 into the equation and solve it for . Which equation describes ? Use the hints below if you get stuck.

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Hints

Part 4: Finding the Limit

Now that we have an expression for in terms of , we can evaluate its limit as approaches zero: Note that direct substitution in this instance fails, resulting in indeterminate form: Our expression for contains a removable discontinuity. There are multiple ways we can address this. Without using calculus, we can evaluate this limit by rationalizing the denominator, using the conjugate method. After rationalizing the denominator and simplifying the result, which of the following represents the same value as the original limit? Use the hints below if you get stuck.

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Hints

Evaluating the Limit

Now that we have removed the discontinuity, we can evaluate the limit and find the result we were looking for. What is the result of evaluating the limit? Use the hints below if you get stuck.

Hints

Part 5: Reflecion

Was your prediction correct? Were you surprised by the results?

Visualizing the Results

Use the slider to adjust the value of the radius, or select "Start Animation."

Part 6: Calculus Extension

Before rationalizing the denominator in our limit expression we had the following:  Another way to evaluate this limit is by using L'Hôpital's Rule. This rule states that to evaluate a limit that results in indeterminate form, differentiate the numerator and differentiate the denominator, and then evaluate the resulting limit. Using L'Hôpital's Rule and the Chain Rule, evaluate the limit. Click "Check" at the bottom of the page to verify all of your results.

Inspiration for this problem is credited to the following source: