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The figure shows a fixed circle $ C_1 $ with equation $ (x - 1)^2 + y^2 = 1 $ and a shrinking circle $ C_2 $ with radius $ r $ and center the origin. $ P $ is the point $ (0,r) $, $ Q $ is the upper point of intersection of the two circles, and $ R $ is the point of intersection of the line $ PQ $ and the $ x $-axis. What happens to $ R $ as $ C_2 $ shrinks, that is, as $ r \to 0^+ $?

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Harvey Mudd College

Baylor University

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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