Chapter 2 | Limits
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2.2 | The Limit of a Function
Learning Objectives 2.2.1 Using correct notation, describe the limit of a function. 2.2.2 Use a table of values to estimate the limit of a function or to identify when the limit does not exist. 2.2.3 Use a graph to estimate the limit of a function or to identify when the limit does not exist. 2.2.4 Define one-sided limits and provide examples. 2.2.5 Explain the relationship between one-sided and two-sided limits. 2.2.6 Using correct notation, describe an infinite limit. 2.2.7 Define a vertical asymptote. The concept of a limit or limiting process, essential to the understanding of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting process to obtain better and better approximations of areas of circles. Yet, the formal definition of a limit—as we know and understand it today—did not appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, by using an intuitive approach. At the end of this chapter, armed with a conceptual understanding of limits, we examine the formal definition of a limit. We begin our exploration of limits by taking a look at the graphs of the functions f ( x ) = x 2 −4 x −2 , g ( x ) = | x −2| x −2 , and h ( x ) = 1 ( x −2) 2 , which are shown in Figure 2.12 . In particular, let’s focus our attention on the behavior of each graph at and around x =2.
Figure 2.12 These graphs show the behavior of three different functions around x =2.
Each of the three functions is undefined at x =2, but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of x =2. To express the behavior of each graph in the vicinity of 2 more completely, we need to introduce the concept of a limit. Intuitive Definition of a Limit Let’s first take a closer look at how the function f ( x ) = ( x 2 −4)/( x −2) behaves around x =2 in Figure 2.12 . As the values of x approach 2 from either side of 2, the values of y = f ( x ) approach 4. Mathematically, we say that the limit of f ( x ) as x approaches 2 is 4. Symbolically, we express this limit as
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