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Intuitive Definition of a Limit

Let’s first take a closer look at how the function \(f \left(\right. x \left.\right) = \left(\right. x^{2} - 4 \left.\right) / \left(\right. x - 2 \left.\right)\) behaves around \(x = 2\) in Figure 2.12. As the values of x approach 2 from either side of 2, the values of \(y = f \left(\right. x \left.\right)\) approach 4. Mathematically, we say that the limit of \(f \left(\right. x \left.\right)\) as x approaches 2 is 4. Symbolically, we express this limit as

\[\underset{x \rightarrow 2}{\text{lim}} f \left(\right. x \left.\right) = 4 .\]

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

\[\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) = L .\]

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 \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) ,\) 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. Table 2.1 demonstrates what your tables might look like.
    x \(f \left(\right. x \left.\right)\) x \(f \left(\right. x \left.\right)\)
    \(a - 0.1\) \(f \left(\right. a - 0.1 \left.\right)\) \(a + 0.1\) \(f \left(\right. a + 0.1 \left.\right)\)
    \(a - 0.01\) \(f \left(\right. a - 0.01 \left.\right)\) \(a + 0.01\) \(f \left(\right. a + 0.01 \left.\right)\)
    \(a - 0.001\) \(f \left(\right. a - 0.001 \left.\right)\) \(a + 0.001\) \(f \left(\right. a + 0.001 \left.\right)\)
    \(a - 0.0001\) \(f \left(\right. a - 0.0001 \left.\right)\) \(a + 0.0001\) \(f \left(\right. a + 0.0001 \left.\right)\)
    Use additional values as necessary. Use additional values as necessary.
    Table 2.1 Table of Functional Values for \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right)\)
  2. Next, let’s look at the values in each of the \(f \left(\right. x \left.\right)\) 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 \(f \left(\right. a - 0.1 \left.\right) , f \left(\right. a - 0.01 \left.\right) , f \left(\right. a - 0.001 \left.\right) . , f \left(\right. a - 0.0001 \left.\right) ,\) and so on, and \(f \left(\right. a + 0.1 \left.\right) , f \left(\right. a + 0.01 \left.\right) , f \left(\right. a + 0.001 \left.\right) , f \left(\right. a + 0.0001 \left.\right) ,\) and so on. (Note: Although we have chosen the x-values \(a \pm 0.1 , a \pm 0.01 , a \pm 0.001 , a \pm 0.0001 ,\) 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 \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) = L .\) 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 to graph functions, we can plot the function \(f \left(\right. x \left.\right) ,\) making sure the functional values of \(f \left(\right. x \left.\right)\) 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 \(\underset{x \rightarrow a}{\text{lim}} f \left(\right. x \left.\right) = L .\) We may need to zoom in on our graph and repeat this process several times.

We apply this Problem-Solving Strategy to compute a limit in Example 2.4.

Example 2.4

Evaluating a Limit Using a Table of Functional Values 1

Evaluate \(\underset{x \rightarrow 0}{\text{lim}} \frac{\text{sin} x}{x}\) using a table of functional values.

Solution

We have calculated the values of \(f \left(\right. x \left.\right) = \left(\right. \text{sin} x \left.\right) / x\) for the values of x listed in Table 2.2.

x \(\frac{\text{sin} x}{x}\) x \(\frac{\text{sin} x}{x}\)
−0.1 0.998334166468 0.1 0.998334166468
−0.01 0.999983333417 0.01 0.999983333417
−0.001 0.999999833333 0.001 0.999999833333
−0.0001 0.999999998333 0.0001 0.999999998333
Table 2.2 Table of Functional Values for \(\underset{x \rightarrow 0}{\text{lim}} \frac{\text{sin} x}{x}\)

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 \(\frac{\left(\right. \text{sin} x \left.\right)}{x}\) column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that \(\underset{x \rightarrow 0}{\text{lim}} \frac{\text{sin} x}{x} = 1 .\) A calculator or computer-generated graph of \(f \left(\right. x \left.\right) = \frac{\left(\right. \text{sin} x \left.\right)}{x}\) would be similar to that shown in Figure 2.13, and it confirms our estimate.

A graph of f(x) = sin(x)/x over the interval [-6, 6]. The curving function has a y intercept at x=0 and x intercepts at y=pi and y=-pi.
Figure 2.13 The graph of \(f \left(\right. x \left.\right) = \left(\right. \text{sin} x \left.\right) / x\) confirms the estimate from Table 2.2.

Example 2.5

Evaluating a Limit Using a Table of Functional Values 2

Evaluate \(\underset{x \rightarrow 4}{\text{lim}} \frac{\sqrt{x} - 2}{x - 4}\) using a table of functional values.

Solution

As before, we use a table—in this case, Table 2.3—to list the values of the function for the given values of x.

x \(\frac{\sqrt{x} - 2}{x - 4}\) x \(\frac{\sqrt{x} - 2}{x - 4}\)
3.9 0.251582341869 4.1 0.248456731317
3.99 0.25015644562 4.01 0.24984394501
3.999 0.250015627 4.001 0.249984377
3.9999 0.250001563 4.0001 0.249998438
3.99999 0.25000016 4.00001 0.24999984
Table 2.3 Table of Functional Values for \(\underset{x \rightarrow 4}{\text{lim}} \frac{\sqrt{x} - 2}{x - 4}\)

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 \(\underset{x \rightarrow 4}{\text{lim}} \frac{\sqrt{x} - 2}{x - 4} = 0.25 .\) We confirm this estimate using the graph of \(f \left(\right. x \left.\right) = \frac{\sqrt{x} - 2}{x - 4}\) shown in Figure 2.14.

A graph of the function f(x) = (sqrt(x) – 2 ) / (x-4) over the interval [0,8]. There is an open circle on the function at x=4. The function curves asymptotically towards the x axis and y axis in quadrant one.
Figure 2.14 The graph of \(f \left(\right. x \left.\right) = \frac{\sqrt{x} - 2}{x - 4}\) confirms the estimate from Table 2.3.

Checkpoint 2.4

Estimate \(\underset{x \rightarrow 1}{\text{lim}} \frac{\frac{1}{x} - 1}{x - 1}\) using a table of functional values. Use a graph to confirm your estimate.

At this point, we see from Example 2.4 and Example 2.5 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 Example 2.6, we evaluate a limit exclusively by looking at a graph rather than by using a table of functional values.

Example 2.6

Evaluating a Limit Using a Graph

For \(g \left(\right. x \left.\right)\) shown in Figure 2.15, evaluate \(\underset{x \rightarrow −1}{\text{lim}} g \left(\right. x \left.\right) .\)

The graph of a generic curving function g(x). In quadrant two, there is an open circle on the function at (-1,3) and a closed circle one unit up at (-1, 4).
Figure 2.15 The graph of \(g \left(\right. x \left.\right)\) includes one value not on a smooth curve.

Solution

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

Based on Example 2.6, 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.

Checkpoint 2.5

Use the graph of \(h \left(\right. x \left.\right)\) in Figure 2.16 to evaluate \(\underset{x \rightarrow 2}{\text{lim}} h \left(\right. x \left.\right) ,\) if possible.

A graph of the function h(x), which is a parabola graphed over [-2.5, 5]. There is an open circle where the vertex should be at the point (2,-1).
Figure 2.16

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.

Theorem 2.1

Two Important Limits

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

  1. \[\underset{x \rightarrow a}{\text{lim}} x = a\]
  2. \[\underset{x \rightarrow a}{\text{lim}} c = c\]

We can make the following observations about these two limits.

  1. For the first limit, observe that as x approaches a, so does \(f \left(\right. x \left.\right) ,\) because \(f \left(\right. x \left.\right) = x .\) Consequently, \(\underset{x \rightarrow a}{\text{lim}} x = a .\)
  2. For the second limit, consider Table 2.4.
x \(f \left(\right. x \left.\right) = c\) x \(f \left(\right. x \left.\right) = c\)
\(a - 0.1\) c \(a + 0.1\) c
\(a - 0.01\) c \(a + 0.01\) c
\(a - 0.001\) c \(a + 0.001\) c
\(a - 0.0001\) c \(a + 0.0001\) c
Table 2.4 Table of Functional Values for \(\underset{x \rightarrow a}{\text{lim}} c = c\)

Observe that for all values of x (regardless of whether they are approaching a), the values \(f \left(\right. x \left.\right)\) remain constant at c. We have no choice but to conclude \(\underset{x \rightarrow a}{\text{lim}} c = c .\)

This lesson is part of:

Limits

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