Учебные пособия > Calculus Volume 1

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

which are shown in [link]. In particular, let’s focus our attention on the behavior of each graph at and around [latex]x=2.[/latex]

Each of the three functions is undefined at [latex]x=2,[/latex] but if we make this statement and no other, we give a very incomplete picture of how each function behaves in the vicinity of [latex]x=2.[/latex] 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 [latex]f\left(x\right)=\left(^-4\right)\text\left(x-2\right)[/latex] behaves around [latex]x=2[/latex] in [link]. As the values of x approach 2 from either side of 2, the values of [latex]y=f\left(x\right)[/latex] approach 4. Mathematically, we say that the limit of [latex]f\left(x\right)[/latex] as x approaches 2 is 4. Symbolically, we express this limit as

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:

Let [latex]f\left(x\right)[/latex] 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 [latex]f\left(x\right)[/latex] approach the real number L as the values of [latex]x\left(\ne a\right)[/latex] approach the number a, then we say that the limit of [latex]f\left(x\right)[/latex] as x approaches a is L. (More succinct, as x gets closer to a, [latex]f\left(x\right)[/latex] gets closer and stays close to L.) Symbolically, we express this idea as

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.

1. To evaluate [latex]\underset>f\left(x\right),[/latex] 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.

We apply this Problem-Solving Strategy to compute a limit in [link].

Evaluate [latex]\underset>\frac\phantom>x>[/latex] using a table of functional values.

We have calculated the values of [latex]f\left(x\right)=\left(\text\phantom>x\right)\textx[/latex] for the values of x listed in [link].

Note: The values in this table were obtained using a calculator and using all the places given in the calculator output.

As we read down each [latex]\frac<\left(\text\phantom>x\right)>[/latex] column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that [latex]\underset>\frac\phantom>x>=1.[/latex] A calculator-or computer-generated graph of [latex]f\left(x\right)=\frac<\left(\text\phantom>x\right)>[/latex] would be similar to that shown in [link], and it confirms our estimate.

The graph of [latex]f\left(x\right)=\left(\text\phantom>x\right)\textx[/latex] confirms the estimate from [link].

Evaluate [latex]\underset>\frac-2>[/latex] using a table of functional values.

As before, we use a table—in this case, [link]—to list the values of the function for the given values of x.

After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that [latex]\underset>\frac-2>=0.25.[/latex] We confirm this estimate using the graph of [latex]f\left(x\right)=\frac-2>[/latex] shown in [link].

Estimate [latex]\underset>\frac-1>[/latex] using a table of functional values. Use a graph to confirm your estimate.

Use 0.9, 0.99, 0.999, 0.9999, 0.99999 and 1.1, 1.01, 1.001, 1.0001, 1.00001 as your table values.

At this point, we see from [link] and [link] that it may be just as easy, if not easier, to estimate a limit of a function by inspecting its graph as it is to estimate the limit by using a table of functional values. In [link], we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.

For [latex]g\left(x\right)[/latex] shown in [link], evaluate [latex]\underset>g\left(x\right).[/latex]

Despite the fact that [latex]g\left(-1\right)=4,[/latex] as the x-values approach −1 from either side, the [latex]g\left(x\right)[/latex] values approach 3. Therefore, [latex]\underset>g\left(x\right)=3.[/latex] Note that we can determine this limit without even knowing the algebraic expression of the function.

Based on [link], we make the following observation: It is possible for the limit of a function to exist at a point, and for the function to be defined at this point, but the limit of the function and the value of the function at the point may be different.

Use the graph of [latex]h\left(x\right)[/latex] in [link] to evaluate [latex]\underset>h\left(x\right),[/latex] if possible.

What y-value does the function approach as the x-values approach 2?

Looking at a table of functional values or looking at the graph of a function provides us with useful insight into the value of the limit of a function at a given point. However, these techniques rely too much on guesswork. We eventually need to develop alternative methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the next section; however, at this point we introduce two special limits that are foundational to the techniques to come.

Let a be a real number and c be a constant.

We can make the following observations about these two limits.

1. For the first limit, observe that as x approaches a, so does [latex]f\left(x\right),[/latex] because [latex]f\left(x\right)=x.[/latex] Consequently, [latex]\underset>x=a.[/latex]
2. For the second limit, consider [link].

Observe that for all values of x (regardless of whether they are approaching a), the values [latex]f\left(x\right)[/latex] remain constant at c. We have no choice but to conclude [latex]\underset>c=c.[/latex]

The Existence of a Limit

As we consider the limit in the next example, keep in mind that for the limit of a function to exist at a point, the functional values must approach a single real-number value at that point. If the functional values do not approach a single value, then the limit does not exist.

Evaluate [latex]\underset>\text\phantom>\left(1\text\mathit>\right)[/latex] using a table of values.

[link] lists values for the function [latex]\text\left(1\textx\right)[/latex] for the given values of x.

After examining the table of functional values, we can see that the y-values do not seem to approach any one single value. It appears the limit does not exist. Before drawing this conclusion, let’s take a more systematic approach. Take the following sequence of x-values approaching 0:

The corresponding y-values are

At this point we can indeed conclude that [latex]\underset>\text\phantom>\left(1\text\mathit>\right)[/latex] does not exist. (Mathematicians frequently abbreviate “does not exist” as DNE. Thus, we would write [latex]\underset>\text\phantom>\left(1\text\mathit>\right)[/latex] DNE.) The graph of [latex]f\left(x\right)=\text\phantom>\left(1\textx\right)[/latex] is shown in [link] and it gives a clearer picture of the behavior of [latex]\text\left(1\textx\right)[/latex] as x approaches 0. You can see that [latex]\text\left(1\text\mathit>\right)[/latex] oscillates ever more wildly between −1 and 1 as x approaches 0.

The graph of [latex]f\left(x\right)=\text\left(1\textx\right)[/latex] oscillates rapidly between −1 and 1 as x approaches 0.

The oscillations are less frequent as the function moves away from 0 on the x axis." />

Use a table of functional values to evaluate [latex]\underset>\frac<|^-4|>,[/latex] if possible.

Use x-values 1.9, 1.99, 1.999, 1.9999, 1.9999 and 2.1, 2.01, 2.001, 2.0001, 2.00001 in your table.

One-Sided Limits

Sometimes indicating that the limit of a function fails to exist at a point does not provide us with enough information about the behavior of the function at that particular point. To see this, we now revisit the function [latex]g\left(x\right)=|x-2|\text\left(x-2\right)[/latex] introduced at the beginning of the section (see [link](b)). As we pick values of x close to 2, [latex]g\left(x\right)[/latex] does not approach a single value, so the limit as x approaches 2 does not exist—that is, [latex]\underset>g\left(x\right)[/latex] DNE. However, this statement alone does not give us a complete picture of the behavior of the function around the x-value 2. To provide a more accurate description, we introduce the idea of a one-sided limit . For all values to the left of 2 (or the negative side of 2), [latex]g\left(x\right)=-1.[/latex] Thus, as x approaches 2 from the left, [latex]g\left(x\right)[/latex] approaches −1. Mathematically, we say that the limit as x approaches 2 from the left is −1. Symbolically, we express this idea as

Similarly, as x approaches 2 from the right (or from the positive side), [latex]g\left(x\right)[/latex] approaches 1. Symbolically, we express this idea as

We can now present an informal definition of one-sided limits.

We define two types of one-sided limits.

Limit from the left: Let [latex]f\left(x\right)[/latex] be a function defined at all values in an open interval of the form z, and let L be a real number. If the values of the function [latex]f\left(x\right)[/latex] approach the real number L as the values of x (where [latex]x<\mathit<\text>\text<)>[/latex] approach the number a, then we say that L is the limit of [latex]f\left(x\right)[/latex] as x approaches a from the left. Symbolically, we express this idea as

Limit from the right: Let [latex]f\left(x\right)[/latex] be a function defined at all values in an open interval of the form [latex]\left(a,c\right),[/latex] and let L be a real number. If the values of the function [latex]f\left(x\right)[/latex] approach the real number L as the values of x (where [latex]x>\mathit\text<)>[/latex] approach the number a, then we say that L is the limit of [latex]f\left(x\right)[/latex] as x approaches a from the right. Symbolically, we express this idea as

For the function [latex]f\left(x\right)=\left\x+1& \text\phantom>x<2\hfill \\ ^-4& \text\phantom>x\ge 2\hfill \end,[/latex] evaluate each of the following limits.

1. [latex]\underset^>>f\left(x\right)[/latex]
2. [latex]\underset^>>f\left(x\right)[/latex]

We can use tables of functional values again [link]. Observe that for values of x less than 2, we use [latex]f\left(x\right)=x+1[/latex] and for values of x greater than 2, we use [latex]f\left(x\right)=^-4.[/latex]

Based on this table, we can conclude that a. [latex]\underset^>>f\left(x\right)=3[/latex] and b. [latex]\underset^>>f\left(x\right)=0.[/latex] Therefore, the (two-sided) limit of [latex]f\left(x\right)[/latex] does not exist at [latex]x=2.[/latex] [link] shows a graph of [latex]f\left(x\right)[/latex] and reinforces our conclusion about these limits.

Use a table of functional values to estimate the following limits, if possible.