The Limit of a Function

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

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)

in [link]. As the values of x approach 2 from either side of 2, the values of

y = f (x)

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:

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.

Table of Functional Values for $lim_{x \to a} f (x)$
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.

Table of Functional Values for $lim_{x \to 0} \frac{sin x}{x}$
x	$\frac{sin x}{x}$		x	$\frac{sin x}{x}$
−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 of Functional Values for $lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}$
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

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.

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.

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.

Table of Functional Values for $lim_{x \to a} c = c$
x	$f (x) = c$		x	$f (x) = c$
$a - 0.1$	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

Observe that for all values of x (regardless of whether they are approaching a), the values

f (x)

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.

One-Sided Limits

Table of Functional Values for $lim_{x \to 0} sin (\frac{1}{x})$
x	$sin (\frac{1}{x})$		x	$sin (\frac{1}{x})$
−0.1	0.544021110889		0.1	−0.544021110889
−0.01	0.50636564111	0.01	−0.50636564111
−0.001	−0.8268795405312	0.001	0.826879540532
−0.0001	0.305614388888	0.0001	−0.305614388888
−0.00001	−0.035748797987	0.00001	0.035748797987
−0.000001	0.349993504187	0.000001	−0.349993504187

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

g (x) = \| x - 2 \| / (x - 2)

introduced at the beginning of the section (see [link](b)). As we pick values of x close to 2,

g (x)

does not approach a single value, so the limit as x approaches 2 does not exist—that is,

lim_{x \to 2} g (x)

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),

g (x) = −1 .

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),

g (x)

Table of Functional Values for $f (x) = {\begin{cases} x + 1 if x < 2 \\ x^{2} - 4 if x \geq 2 \end{cases}$
x	$f (x) = x + 1$		x	$f (x) = x^{2} −4$
1.9	2.9		2.1	0.41
1.99	2.99	2.01	0.0401
1.999	2.999	2.001	0.004001
1.9999	2.9999	2.0001	0.00040001
1.99999	2.99999	2.00001	0.0000400001

Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist. These conclusions are summarized in [link].

Infinite Limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

the third and final function introduced at the beginning of this section (see [link](c)). From its graph we see that as the values of x approach 2, the values of

h (x) = 1 / {(x - 2)}^{2}

become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of

h (x)

It is important to understand that when we write statements such as

lim_{x \to a} f (x) = + \infty

we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function

f (x)

to exist at a, it must approach a real number L as x approaches a. That said, if, for example,

lim_{x \to a} f (x) = + \infty,

Table of Functional Values for $f (x) = \frac{1}{x}$
x	$\frac{1}{x}$		x	$\frac{1}{x}$
−0.1	−10		0.1	10
−0.01	−100	0.01	100
−0.001	−1000	0.001	1000
−0.0001	−10,000	0.0001	10,000
−0.00001	−100,000	0.00001	100,000
−0.000001	−1,000,000	0.000001	1,000,000

It is useful to point out that functions of the form

f (x) = 1 / {(x - a)}^{n},

where n is a positive integer, have infinite limits as x approaches a from either the left or right ([link]). These limits are summarized in [link].

points on the graph having x-coordinates very near to a are very close to the vertical line

x = a .

is called a vertical asymptote of the graph. We formally define a vertical asymptote as follows:

In the next example we put our knowledge of various types of limits to use to analyze the behavior of a function at several different points.

Key Concepts

Key Equations

Ratio of Masses and Speeds for a Moving Object
$\frac{v}{c}$	$\sqrt{1 - \frac{v^{2}}{c^{2}}}$	$\frac{m}{m_{0}}$
0.99	0.1411	7.089
0.999	0.0447	22.37
0.9999	0.0141	70.71

For the following exercises, consider the function $f (x) = \frac{x^{2} - 1}{\| x - 1 \|} .$

[T] Complete the following table for the function. Round your solutions to four decimal places.

f (x)

f (x)

| {: valign=”top”}|———- | 0.9 | a. | 1.1 | e. | {: valign=”top”}| 0.99 | b. | 1.01 | f. | {: valign=”top”}| 0.999 | c. | 1.001 | g. | {: valign=”top”}| 0.9999 | d. | 1.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, f(x), x, and f(x). The values of the first column under the header are 0.9, .99, 0.999, and 0.9999. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 1.1, 1.01, 1.001, and 1.0001. The values of the fourth column under the header are e, f, g, and h.”}

What do your results in the preceding exercise indicate about the two-sided limit $lim_{x \to 1} f (x) ?$

Explain your response.

lim_{x \to 1} f (x)

does not exist because $lim_{x \to 1^{-}} f (x) = −2 \neq lim_{x \to 1^{+}} f (x) = 2 .$

For the following exercises, consider the function $f (x) = {(1 + x)}^{1 / x} .$

[T] Make a table showing the values of f for $x = −0.01, −0.001, −0.0001, −0.00001$

and for $x = 0.01, 0.001, 0.0001, 0.00001 .$

Round your solutions to five decimal places.

f (x)

f (x)

| {: valign=”top”}|———- | −0.01 | a. | 0.01 | e. | {: valign=”top”}| −0.001 | b. | 0.001 | f. | {: valign=”top”}| −0.0001 | c. | 0.0001 | g. | {: valign=”top”}| −0.00001 | d. | 0.00001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, f(x), x, and f(x). The values of the first column under the header are -0.01, -0.001, -0.0001, and -0.00001. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 0.01, 0.001, 0.0001, and 0.00001. The values of the fourth column under the header are e, f, g, and h.”}

What does the table of values in the preceding exercise indicate about the function $f (x) = {(1 + x)}^{1 / x} ?$

lim_{x \to 0} {(1 + x)}^{1 / x} = 2.7183

To which mathematical constant does the limit in the preceding exercise appear to be getting closer?

In the following exercises, use the given values to set up a table to evaluate the limits. Round your solutions to eight decimal places.

[T] $lim_{x \to 0} \frac{sin 2 x}{x}; ±0.1, ±0.01, ±0.001, ±.0001$

x	$\frac{sin 2 x}{x}$

x	$\frac{sin 2 x}{x}$

| {: valign=”top”}|———- | −0.1 | a. | 0.1 | e. | {: valign=”top”}| −0.01 | b. | 0.01 | f. | {: valign=”top”}| −0.001 | c. | 0.001 | g. | {: valign=”top”}| −0.0001 | d. | 0.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, sin(2x)/x, x, and sin(2x) / x. The values of the first column under the header are -0.1, -0.01, -0.001, and -0.0001. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the fourth column under the header are e, f, g, and h.”}

a. 1.98669331; b. 1.99986667; c. 1.99999867; d. 1.99999999; e. 1.98669331; f. 1.99986667; g. 1.99999867; h. 1.99999999; $lim_{x \to 0} \frac{sin 2 x}{x} = 2$

[T] $lim_{x \to 0} \frac{sin 3 x}{x}$

±0.1, ±0.01, ±0.001, ±0.0001

X	$\frac{sin 3 x}{x}$

x	$\frac{sin 3 x}{x}$

| {: valign=”top”}|———- | −0.1 | a. | 0.1 | e. | {: valign=”top”}| −0.01 | b. | 0.01 | f. | {: valign=”top”}| −0.001 | c. | 0.001 | g. | {: valign=”top”}| −0.0001 | d. | 0.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, sin(3x)/x, x, and sin(3x) / x. The values of the first column under the header are -0.1, -0.01, -0.001, and -0.0001. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the fourth column under the header are e, f, g, and h.”}

Use the preceding two exercises to conjecture (guess) the value of the following limit: $lim_{x \to 0} \frac{sin a x}{x}$

for a, a positive real value.

lim_{x \to 0} \frac{sin a x}{x} = a

[T] In the following exercises, set up a table of values to find the indicated limit. Round to eight digits.

lim_{x \to 2} \frac{x^{2} - 4}{x^{2} + x - 6}

x	$\frac{x^{2} - 4}{x^{2} + x - 6}$

x	$\frac{x^{2} - 4}{x^{2} + x - 6}$

| {: valign=”top”}|———- | 1.9 | a. | 2.1 | e. | {: valign=”top”}| 1.99 | b. | 2.01 | f. | {: valign=”top”}| 1.999 | c. | 2.001 | g. | {: valign=”top”}| 1.9999 | d. | 2.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, (x^2 – 4) / (x^2 + x – 6), x, and (x^2 – 4) / (x^2 + x – 6). The values of the first column under the header are 1.9, 1.99, 1.999, and 1.9999. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 2.1, 2.01, 2.001, and 2.0001. The values of the fourth column under the header are e, f, g, and h.”}

lim_{x \to 1} (1 - 2 x)

1 - 2 x

1 - 2 x

| {: valign=”top”}|———- | 0.9 | a. | 1.1 | e. | {: valign=”top”}| 0.99 | b. | 1.01 | f. | {: valign=”top”}| 0.999 | c. | 1.001 | g. | {: valign=”top”}| 0.9999 | d. | 1.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, 1-2x, x, and 1-2x. The values of the first column under the header are 0.9, 0.99, 0.999, and 0.9999. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 1.1, 1.01, 1.001, and 1.0001. The values of the fourth column under the header are e, f, g, and h.”}

a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −1.0200000; g. −1.0020000; h. −1.0002000; $lim_{x \to 1} (1 - 2 x) = −1$

lim_{x \to 0} \frac{5}{1 - e^{1 / x}}

x	$\frac{5}{1 - e^{1 / x}}$

x	$\frac{5}{1 - e^{1 / x}}$

| {: valign=”top”}|———- | −0.1 | a. | 0.1 | e. | {: valign=”top”}| −0.01 | b. | 0.01 | f. | {: valign=”top”}| −0.001 | c. | 0.001 | g. | {: valign=”top”}| −0.0001 | d. | 0.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, 5 / (1 – e^ (1/x) ), x, and 5 / (1 – e^ (1/x) ). The values of the first column under the header are -0.1, -0.01, -0.001, and -0.0001. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the fourth column under the header are e, f, g, and h.”}

lim_{z \to 0} \frac{z - 1}{z^{2} (z + 3)}

z	$\frac{z - 1}{z^{2} (z + 3)}$

z	$\frac{z - 1}{z^{2} (z + 3)}$

| {: valign=”top”}|———- | −0.1 | a. | 0.1 | e. | {: valign=”top”}| −0.01 | b. | 0.01 | f. | {: valign=”top”}| −0.001 | c. | 0.001 | g. | {: valign=”top”}| −0.0001 | d. | 0.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings z, (z-1) / ((z^2)(z+3)), z, and (z-1) / ((z^2)(z+3)). The values of the first column under the header are -0.1, -0.01, -0.001, and -0.0001. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the fourth column under the header are e, f, g, and h.”}

a. −37.931934; b. −3377.9264; c. −333,777.93; d. −33,337,778; e. −29.032258; f. −3289.0365; g. −332,889.04; h. −33,328,889 $lim_{x \to 0} \frac{z - 1}{z^{2} (z + 3)} = − \infty$

lim_{t \to 0^{+}} \frac{cos t}{t}

t	$\frac{cos t}{t}$

| {: valign=”top”}|———- | 0.1 | a. | {: valign=”top”}| 0.01 | b. | {: valign=”top”}| 0.001 | c. | {: valign=”top”}| 0.0001 | d. | {: valign=”top”}{: .unnumbered summary=”A table with two columns and five rows. The first row contains the headings t and cos(t) / t. The values of the first column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the second column under the header are a, b, c, and d.”}

lim_{x \to 2} \frac{1 - \frac{2}{x}}{x^{2} - 4}

x	$\frac{1 - \frac{2}{x}}{x^{2} - 4}$

x	$\frac{1 - \frac{2}{x}}{x^{2} - 4}$

| {: valign=”top”}|———- | 1.9 | a. | 2.1 | e. | {: valign=”top”}| 1.99 | b. | 2.01 | f. | {: valign=”top”}| 1.999 | c. | 2.001 | g. | {: valign=”top”}| 1.9999 | d. | 2.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings x, (1- (2/x)) / (x^2 – 4 ), x, and (1-(2/x)) / (x^2 – 4). The values of the first column under the header are 1.9, 1.99, 1.999, and 1.9999. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 2.1, 2.01, 2.001, and 2.0001. The values of the fourth column under the header are e, f, g, and h.”}

a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; e. 0.11614402; f. 0.12406794; g. 0.12490631; h. 0.12499063; $∴ lim_{x \to 2} \frac{1 - \frac{2}{x}}{x^{2} - 4} = 0.1250 = \frac{1}{8}$

[T] In the following exercises, set up a table of values and round to eight significant digits. Based on the table of values, make a guess about what the limit is. Then, use a calculator to graph the function and determine the limit. Was the conjecture correct? If not, why does the method of tables fail?

lim_{θ \to 0} sin (\frac{π}{θ})

θ	$sin (\frac{π}{θ})$

θ	$sin (\frac{π}{θ})$

| {: valign=”top”}|———- | −0.1 | a. | 0.1 | e. | {: valign=”top”}| −0.01 | b. | 0.01 | f. | {: valign=”top”}| −0.001 | c. | 0.001 | g. | {: valign=”top”}| −0.0001 | d. | 0.0001 | h. | {: valign=”top”}{: .unnumbered summary=”A table with four columns and five rows. The first row contains the headings theta, sin(pi/theta), theta, sin(pi/theta). The values of the first column under the header are -0.1, -0.01, -0.001, and -0.0001. The values of the second column under the header are a, b, c, and d. The values of the third column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the fourth column under the header are e, f, g, and h.”}

lim_{α \to 0^{+}} \frac{1}{α} cos (\frac{π}{α})

a	$\frac{1}{α} cos (\frac{π}{α})$

| {: valign=”top”}|———- | 0.1 | a. | {: valign=”top”}| 0.01 | b. | {: valign=”top”}| 0.001 | c. | {: valign=”top”}| 0.0001 | d. | {: valign=”top”}{: .unnumbered summary=”A table with two columns and five rows. The first row contains the headings A and (1/alpha) * cos(pi/alpha). The values of the first column under the header are 0.1, 0.01, 0.001, and 0.0001. The values of the second column under the header are a, b, c, and d.”}

a. −10.00000; b. −100.00000; c. −1000.0000; d. −10,000.000; Guess: $lim_{α \to 0^{+}} \frac{1}{α} cos (\frac{π}{α}) = \infty,$

actual: DNE* * *

A graph of the function (1/alpha) * cos (pi / alpha), which oscillates gently until the interval [-.2, .2], where it oscillates rapidly, going to infinity and negative infinity as it approaches the y axis.

In the following exercises, consider the graph of the function $y = f (x)$

shown here. Which of the statements about $y = f (x)$

are true and which are false? Explain why a statement is false.

lim_{x \to 10} f (x) = 0

lim_{x \to {−2}^{+}} f (x) = 3

False; $lim_{x \to {−2}^{+}} f (x) = + \infty$

lim_{x \to −8} f (x) = f (−8)

lim_{x \to 6} f (x) = 5

False; $lim_{x \to 6} f (x)$

DNE since $lim_{x \to 6^{-}} f (x) = 2$

and $lim_{x \to 6^{+}} f (x) = 5 .$

In the following exercises, use the following graph of the function $y = f (x)$

to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first segment exists for x <=1, and the second segment exists for x > 1. The first segment is linear with a slope of 1 and goes through the origin. Its endpoint is a closed circle at (1,1). The second segment is also linear with a slope of -1. It begins with the open circle at (1,2).

lim_{x \to 1^{-}} f (x)

lim_{x \to 1^{+}} f (x)

lim_{x \to 1} f (x)

lim_{x \to 2} f (x)

f (1)

In the following exercises, use the graph of the function $y = f (x)$

shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first is a linear function for x < 0. There is an open circle at (0,1), and its slope is -1. The second segment is the right half of a parabola opening upward. Its vertex is a closed circle at (0, -4), and it goes through the point (2,0).

lim_{x \to 0^{-}} f (x)

lim_{x \to 0^{+}} f (x)

lim_{x \to 0} f (x)

DNE

lim_{x \to 2} f (x)

In the following exercises, use the graph of the function $y = f (x)$

shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with three segments, all linear. The first exists for x < -2, has a slope of 1, and ends at the open circle at (-2, 0). The second exists over the interval [-2, 2], has a slope of -1, goes through the origin, and has closed circles at its endpoints (-2, 2) and (2,-2). The third exists for x>2, has a slope of 1, and begins at the open circle (2,2).

lim_{x \to {−2}^{-}} f (x)

lim_{x \to {−2}^{+}} f (x)

lim_{x \to −2} f (x)

DNE

lim_{x \to 2^{-}} f (x)

lim_{x \to 2^{+}} f (x)

lim_{x \to 2} f (x)

In the following exercises, use the graph of the function $y = g (x)$

shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The first exists for x>=0 and is the left half of an upward opening parabola with vertex at the closed circle (0,3). The second exists for x>0 and is the right half of a downward opening parabola with vertex at the open circle (0,0).

lim_{x \to 0^{-}} g (x)

lim_{x \to 0^{+}} g (x)

lim_{x \to 0} g (x)

DNE

In the following exercises, use the graph of the function $y = h (x)$

shown here to find the values, if possible. Estimate when necessary.

A graph of a function with two curves approaching 0 from quadrant 1 and quadrant 3. The curve in quadrant one appears to be the top half of a parabola opening to the right of the y axis along the x axis with vertex at the origin. The curve in quadrant three appears to be the left half of a parabola opening downward with vertex at the origin.

lim_{x \to 0^{-}} h (x)

lim_{x \to 0^{+}} h (x)

lim_{x \to 0} h (x)

In the following exercises, use the graph of the function $y = f (x)$

shown here to find the values, if possible. Estimate when necessary.

A graph with a curve and a point. The point is a closed circle at (0,-2). The curve is part of an upward opening parabola with vertex at (1,-1). It exists for x > 0, and there is a closed circle at the origin.

lim_{x \to 0^{-}} f (x)

−2

lim_{x \to 0^{+}} f (x)

lim_{x \to 0} f (x)

DNE

lim_{x \to 1} f (x)

lim_{x \to 2} f (x)

In the following exercises, sketch the graph of a function with the given properties.

lim_{x \to 2} f (x) = 1, lim_{x \to 4^{-}} f (x) = 3, lim_{x \to 4^{+}} f (x) = 6, x = 4

is not defined.

lim_{x \to - \infty} f (x) = 0, lim_{x \to {−1}^{-}} f (x) = − \infty,

lim_{x \to {−1}^{+}} f (x) = \infty, lim_{x \to 0} f (x) = f (0), f (0) = 1, lim_{x \to \infty} f (x) = − \infty

Answers may vary.* * *

lim_{x \to - \infty} f (x) = 2, lim_{x \to 3^{-}} f (x) = − \infty,

lim_{x \to 3^{+}} f (x) = \infty, lim_{x \to \infty} f (x) = 2, f (0) = \frac{−1}{3}

lim_{x \to - \infty} f (x) = 2, lim_{x \to −2} f (x) = − \infty,

lim_{x \to \infty} f (x) = 2, f (0) = 0

Answers may vary.* * *

A graph containing two curves. The first goes to 2 asymptotically along y=2 and to negative infinity along x = -2. The second goes to negative infinity along x=-2 and to 2 along y=2.

lim_{x \to - \infty} f (x) = 0, lim_{x \to {−1}^{-}} f (x) = \infty, lim_{x \to {−1}^{+}} f (x) = − \infty,

f (0) = −1, lim_{x \to 1^{-}} f (x) = − \infty, lim_{x \to 1^{+}} f (x) = \infty, lim_{x \to \infty} f (x) = 0

Shock waves arise in many physical applications, ranging from supernovas to detonation waves. A graph of the density of a shock wave with respect to distance, x, is shown here. We are mainly interested in the location of the front of the shock, labeled $x_{SF}$

in the diagram.

A graph in quadrant one of the density of a shockwave with three labeled points: p1 and p2 on the y axis, with p1 > p2, and xsf on the x axis. It consists of y= p1 from 0 to xsf, x = xsf from y= p1 to y=p2, and y=p2 for values greater than or equal to xsf.

Evaluate $lim_{x \to x_{S F}^{+}} ρ (x) .$
Evaluate $lim_{x \to x_{S F}^{-}} ρ (x) .$
Evaluate $lim_{x \to x_{S F}} ρ (x) .$
Explain the physical meanings behind your answers.

a. $ρ_{2}$

b. $ρ_{1}$

c. DNE unless $ρ_{1} = ρ_{2} .$

As you approach $x_{SF}$

from the right, you are in the high-density area of the shock. When you approach from the left, you have not experienced the “shock” yet and are at a lower density.

A track coach uses a camera with a fast shutter to estimate the position of a runner with respect to time. A table of the values of position of the athlete versus time is given here, where x is the position in meters of the runner and t is time in seconds. What is $lim_{t \to 2} x (t) ?$

What does it mean physically?

| t (sec) | x (m) | {: valign=”top”}|———- | 1.75 | 4.5 | {: valign=”top”}| 1.95 | 6.1 | {: valign=”top”}| 1.99 | 6.42 | {: valign=”top”}| 2.01 | 6.58 | {: valign=”top”}| 2.05 | 6.9 | {: valign=”top”}| 2.25 | 8.5 | {: valign=”top”}{: .unnumbered summary=”A table with two columns and seven rows. The first row contains the headings t (sec) and x (m). The values of the first column under the header are 1.75, 1.95, 1.99, 2.01, 2.05, and 2.25. The values of the second column under the header are 4.5, 6.1, 6.42, 6.58, 6.9, and 8.5.”}

The Limit of a Function

Intuitive Definition of a Limit

The Existence of a Limit

One-Sided Limits

Infinite Limits

Key Concepts

Key Equations

Glossary