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Chapter 2 | Limits
where m 0 is the object’s mass at rest, v is its speed, and c is the speed of light. What is this speed limit? (We explore this problem further in Example 2.12 .) The idea of a limit is central to all of calculus. We begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. This chapter has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit. 2.1 | A Preview of Calculus Learning Objectives 2.1.1 Describe the tangent problem and how it led to the idea of a derivative. 2.1.2 Explain how the idea of a limit is involved in solving the tangent problem. 2.1.3 Recognize a tangent to a curve at a point as the limit of secant lines. 2.1.4 Identify instantaneous velocity as the limit of average velocity over a small time interval. 2.1.5 Describe the area problem and how it was solved by the integral. 2.1.6 Explain how the idea of a limit is involved in solving the area problem. 2.1.7 Recognize how the ideas of limit, derivative, and integral led to the studies of infinite series and multivariable calculus. As we embark on our study of calculus, we shall see how its development arose from common solutions to practical problems in areas such as engineering physics—like the space travel problem posed in the chapter opener. Two key problems led to the initial formulation of calculus: (1) the tangent problem, or how to determine the slope of a line tangent to a curve at a point; and (2) the area problem, or how to determine the area under a curve. The Tangent Problem and Differential Calculus Rate of change is one of the most critical concepts in calculus. We begin our investigation of rates of change by looking at the graphs of the three lines f ( x ) =−2 x −3, g ( x ) = 1 2 x +1, and h ( x ) =2, shown in Figure 2.2 .
Figure 2.2 The rate of change of a linear function is constant in each of these three graphs, with the constant determined by the slope.
As we move from left to right along the graph of f ( x ) =−2 x −3, we see that the graph decreases at a constant rate. For every 1 unit we move to the right along the x -axis, the y -coordinate decreases by 2 units. This rate of change is determined by the slope (−2) of the line. Similarly, the slope of 1/2 in the function g ( x ) tells us that for every change in x of 1 unit there is a corresponding change in y of 1/2 unit. The function h ( x ) =2 has a slope of zero, indicating that the values of the function remain constant. We see that the slope of each linear function indicates the rate of change of the function.
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