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  • Using correct notation, describe the limit of a function.
  • Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
  • Use a graph to estimate the limit of a function or to identify when the limit does not exist.
  • Define one-sided limits and provide examples.
  • Explain the relationship between one-sided and two-sided limits.
  • Using correct notation, describe an infinite limit.
  • 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 [link] . In particular, let’s focus our attention on the behavior of each graph at and around x = 2 .

Three graphs of functions.  The first is f(s) = (x^2 – 4) / (x-2), which is a line of slope, x intercept (-2,0), and open circle at (2,4). The second is g(x) = |x – 2 | / (x-2), which contains two lines: x=1 for x>2 and x= -1 for x < 2. There are open circles at both endpoints (2, 1) and (-2, 1). The third is h(x) = 1 / (x-2)^2, in which the function curves asymptotically towards y=0 and x=2 in quadrants one and two.
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 [link] . 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

lim x 2 f ( x ) = 4 .

From this very brief informal look at one limit, let’s start to develop an intuitive definition of the limit    . We can think of the limit of a function at a number a as being the one real number L that the functional values approach as the x -values approach a, provided such a real number L exists. Stated more carefully, we have the following definition:

Definition

Let f ( x ) be a function defined at all values in an open interval containing a , with the possible exception of a itself, and let L be a real number. If all values of the function f ( x ) approach the real number L as the values of x ( a ) approach the number a , then we say that the limit of f ( x ) as x approaches a is L . (More succinct, as x gets closer to a , f ( x ) gets closer and stays close to L .) Symbolically, we express this idea as

lim x a f ( x ) = L .

We can estimate limits by constructing tables of functional values and by looking at their graphs. This process is described in the following Problem-Solving Strategy.

Problem-solving strategy: evaluating a limit using a table of functional values

  1. To evaluate lim x a f ( x ) , we begin by completing a table of functional values. We should choose two sets of x -values—one set of values approaching a and less than a , and another set of values approaching a and greater than a . [link] demonstrates what your tables might look like.
    Table of functional values for lim x a f ( x )
    x f ( x ) x f ( x )
    a 0.1 f ( a 0.1 ) a + 0.1 f ( a + 0.1 )
    a 0.01 f ( a 0.01 ) a + 0.01 f ( a + 0.01 )
    a 0.001 f ( a 0.001 ) a + 0.001 f ( a + 0.001 )
    a 0.0001 f ( a 0.0001 ) a + 0.0001 f ( a + 0.0001 )
    Use additional values as necessary. Use additional values as necessary.
  2. Next, let’s look at the values in each of the f ( x ) columns and determine whether the values seem to be approaching a single value as we move down each column. In our columns, we look at the sequence f ( a 0.1 ) , f ( a 0.01 ) , f ( a 0.001 ) . , f ( a 0.0001 ) , and so on, and f ( a + 0.1 ) , f ( a + 0.01 ) , f ( a + 0.001 ) , f ( a + 0.0001 ) , and so on. ( Note : Although we have chosen the x -values a ± 0.1 , a ± 0.01 , a ± 0.001 , a ± 0.0001 , and so forth, and these values will probably work nearly every time, on very rare occasions we may need to modify our choices.)
  3. If both columns approach a common y -value L , we state lim x a f ( x ) = L . We can use the following strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.
  4. Using a graphing calculator or computer software that allows us graph functions, we can plot the function f ( x ) , making sure the functional values of f ( x ) for x -values near a are in our window. We can use the trace feature to move along the graph of the function and watch the y -value readout as the x -values approach a . If the y -values approach L as our x -values approach a from both directions, then lim x a f ( x ) = L . We may need to zoom in on our graph and repeat this process several times.
Practice Key Terms 4

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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