Refer Again to the Graph Figure 1 Select All True Statements About the Graph

2. Limits

2.ii The Limit of a Function

Learning Objectives

Using correct notation, depict the limit of a function.
Use a table of values to estimate the limit of a function or to identify when the limit does non exist.
Use a graph to guess the limit of a part or to identify when the limit does non be.
Define ane-sided limits and provide examples.
Explain the relationship between one-sided and 2-sided limits.
Using correct notation, depict an infinite limit.
Define a vertical asymptote.

The concept of a limit or limiting process, essential to the agreement of calculus, has been around for thousands of years. In fact, early mathematicians used a limiting procedure to obtain better and ameliorate approximations of areas of circles. Yet, the formal definition of a limit—as we know and sympathize it today—did non appear until the late 19th century. We therefore begin our quest to understand limits, as our mathematical ancestors did, past using an intuitive approach. At the terminate of this chapter, armed with a conceptual understanding of limits, we examine the formal definition of a limit.

Nosotros begin our exploration of limits past taking a look at the graphs of the functions

$f(x)=\frac{x^2-4}{x-2}, \, g(x)=\frac{|x-2|}{x-2}$ , and $h(x)=\frac{1}{(x-2)^2}$ ,

which are shown in (Figure). In particular, let's focus our attention on the behavior of each graph at and effectually $x=2$ .

Each of the three functions is undefined at $x=2$ , but if nosotros brand this statement and no other, we requite 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 innovate the concept of a limit.

Intuitive Definition of a Limit

Permit's first take a closer look at how the function $f(x)=(x^2-4)/(x-2)$ behaves around $x=2$ in (Effigy). As the values of $x$ approach ii from either side of 2, the values of $y=f(x)$ approach 4. Mathematically, nosotros say that the limit of $f(x)$ equally $x$ approaches 2 is 4. Symbolically, we express this limit as

$\underset{x \to 2}{\lim}f(x)=4$ .

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

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

Problem-Solving Strategy: Evaluating a Limit Using a Table of Functional Values

To evaluate $\underset{x\to a}{\lim}f(x)$ , we begin by completing a table of functional values. We should cull 2 sets of $x$ -values—one set of values approaching $a$ and less than $a$ , and another set of values budgeted $a$ and greater than $a$ . (Effigy) demonstrates what your tables might look like.

Table of Functional Values for $\underset{x\to a}{\lim}f(x)$
$x$	$f(x)$	$x$	$f(x)$
$a-0.1$	$f(a-0.1)$	$a+0.1$	$f(a+0.1)$
$a-0.01$	$f(a-0.01)$	$a+0.01$	$f(a+0.01)$
$a-0.001$	$f(a-0.001)$	$a+0.001$	$f(a+0.001)$
$a-0.0001$	$f(a-0.0001)$	$a+0.0001$	$f(a+0.0001)$
Use additional values equally necessary.		Utilize boosted values every bit necessary.

Next, let'due south look at the values in each of the $f(x)$ columns and determine whether the values seem to be budgeted a single value as we motion downwardly 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 then on. (Annotation: Although we accept chosen the $x$ -values $a \pm 0.1, \, a \pm 0.01, \, a \pm 0.001, \, a \pm 0.0001$ , and and then forth, and these values will probably piece of work nearly every time, on very rare occasions nosotros may need to modify our choices.)
If both columns approach a common $y$ -value $L$ , we land $\underset{x\to a}{\lim}f(x)=L$ . We can use the post-obit strategy to confirm the result obtained from the table or as an alternative method for estimating a limit.
Using a graphing estimator or computer software that allows us graph functions, we can plot the role $f(x)$ , making sure the functional values of $f(x)$ for $x$ -values most $a$ are in our window. We can use the trace feature to move along the graph of the function and sentry the $y$ -value readout equally the $x$ -values approach $a$ . If the $y$ -values approach $L$ equally our $x$ -values approach $a$ from both directions, then $\underset{x\to a}{\lim}f(x)=L$ . We may need to zoom in on our graph and echo this process several times.

Nosotros employ this Problem-Solving Strategy to compute a limit in (Effigy).

Evaluating a Limit Using a Table of Functional Values 1

Evaluate $\underset{x\to 0}{\lim}\frac{\sin x}{x}$ using a tabular array of functional values.

Evaluating a Limit Using a Table of Functional Values 2

Evaluate $\underset{x\to 4}{\lim}\frac{\sqrt{x}-2}{x-4}$ using a table of functional values.

Estimate $\underset{x\to 1}{\lim}\frac{\frac{1}{x}-1}{x-1}$ using a table of functional values. Utilise a graph to ostend your estimate.

Solution

$\underset{x\to 1}{\lim}\frac{\frac{1}{x}-1}{x-1}=-1$

At this point, nosotros meet from (Effigy) and (Effigy) that information technology may be only every bit like shooting fish in a barrel, if not easier, to estimate a limit of a role by inspecting its graph equally it is to guess the limit past using a table of functional values. In (Effigy), we evaluate a limit exclusively by looking at a graph rather than by using a tabular array of functional values.

Evaluating a Limit Using a Graph

For $g(x)$ shown in (Effigy), evaluate $\underset{x\to -1}{\lim}g(x)$ .

Based on (Figure), we make the following observation: Information technology is possible for the limit of a function to be at a point, and for the function to exist defined at this point, just the limit of the role and the value of the function at the point may be different.

Utilize the graph of $h(x)$ in (Effigy) to evaluate $\underset{x\to 2}{\lim}h(x)$ , if possible.

Solution

$\underset{x\to 2}{\lim}h(x)=-1$ .

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 part at a given point. However, these techniques rely too much on guesswork. Nosotros somewhen need to develop culling methods of evaluating limits. These new methods are more algebraic in nature and we explore them in the side by side department; however, at this indicate we introduce two special limits that are foundational to the techniques to come.

2 Of import Limits

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

$\underset{x\to a}{\lim}x=a$
$\underset{x\to a}{\lim}c=c$

We can brand the post-obit observations about these two limits.

For the first limit, observe that every bit $x$ approaches $a$ , so does $f(x)$ , because $f(x)=x$ . Consequently, $\underset{x\to a}{\lim}x=a$ .
For the 2d limit, consider (Figure).

Table of Functional Values for $\underset{x\to a}{\lim}c=c$
$x$	$f(x)=c$	$x$	$f(x)=c$
$a-0.1$	$c$	$a+0.1$	$c$
$a-0.01$	$c$	$a+0.01$	$c$
$a-0.001$	$c$	$a+0.001$	$c$
$a-0.0001$	$c$	$a+0.0001$	$c$

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

The Existence of a Limit

As we consider the limit in the next example, proceed in listen that for the limit of a part to be at a bespeak, the functional values must approach a single existent-number value at that bespeak. If the functional values do not arroyo a unmarried value, and then the limit does non be.

Evaluating a Limit That Fails to Exist

Evaluate $\underset{x\to 0}{\lim} \sin (1/x)$ using a table of values.

Use a table of functional values to evaluate $\underset{x\to 2}{\lim}\frac{|x^2-4|}{x-2}$ , if possible.

Solution

$\underset{x\to 2}{\lim}\frac{|x^2-4|}{x-2}$ does non exist.

One-Sided Limits

Sometimes indicating that the limit of a function fails to be at a signal does non provide u.s. with enough data about the behavior of the function at that particular betoken. To see this, we at present revisit the function $g(x)=|x-2|/(x-2)$ introduced at the beginning of the section (encounter (Figure)(b)). As we pick values of $x$ close to two, $g(x)$ does not approach a single value, so the limit as $x$ approaches 2 does not exist—that is, $\underset{x\to 2}{\lim}g(x)$ DNE. Even so, this statement lone does not give united states a consummate flick of the beliefs of the function around the $x$ -value 2. To provide a more accurate clarification, we introduce the idea of a ane-sided limit. For all values to the left of 2 (or the negative side of 2), $g(x)=-1$ . Thus, as $x$ approaches 2 from the left, $g(x)$ approaches −1. Mathematically, nosotros say that the limit equally $x$ approaches 2 from the left is −1. Symbolically, we express this idea as

$\underset{x\to 2^-}{\lim}g(x)=-1$ .

Similarly, equally $x$ approaches 2 from the correct (or from the positive side), $g(x)$ approaches one. Symbolically, we express this thought as

$\underset{x\to 2^+}{\lim}g(x)=1$ .

We tin now present an informal definition of ane-sided limits.

Evaluating One-Sided Limits

Utilise a table of functional values to guess the following limits, if possible.

$\underset{x\to 2^-}{\lim}\frac{|x^2-4|}{x-2}$
$\underset{x\to 2^+}{\lim}\frac{|x^2-4|}{x-2}$

Solution

a. $\underset{x\to 2^-}{\lim}\frac{|x^2-4|}{x-2}=-4$ ; b. $\underset{x\to 2^+}{\lim}\frac{|x^2-4|}{x-2}=4$

Let us now consider the relationship betwixt the limit of a function at a point and the limits from the right and left at that signal. It seems articulate 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 accept on unlike values, the limit of the function does not exist. These conclusions are summarized in (Figure).

Space Limits

Evaluating the limit of a part at a point or evaluating the limit of a office from the right and left at a point helps u.s.a. to narrate the behavior of a office around a given value. Equally we shall see, we can also describe the behavior of functions that do non have finite limits.

We now turn our attention to $h(x)=1/(x-2)^2$ , the third and final role introduced at the first of this section (run across (Effigy)(c)). From its graph we run into that as the values of $x$ arroyo two, the values of $h(x)=1/(x-2)^2$ become larger and larger and, in fact, become space. Mathematically, we say that the limit of $h(x)$ as $x$ approaches ii is positive infinity. Symbolically, we express this thought as

$\underset{x\to 2}{\lim}h(x)=+\infty$ .

More more often than not, nosotros define infinite limits as follows:

It is important to understand that when we write statements such equally $\underset{x\to a}{\lim}f(x)=+\infty$ or $\underset{x\to a}{\lim}f(x)=−\infty$ nosotros are describing the behavior of the function, every bit nosotros have but divers it. Nosotros are not asserting that a limit exists. For the limit of a role $f(x)$ to exist at $a$ , it must approach a real number $L$ as $x$ approaches $a$ . That said, if, for example, $\underset{x\to a}{\lim}f(x)=+\infty$ , nosotros always write $\underset{x\to a}{\lim}f(x)=+\infty$ rather than $\underset{x\to a}{\lim}f(x)$ DNE.

Recognizing an Space Limit

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 ((Figure)). These limits are summarized in (Effigy).

Infinite Limits from Positive Integers

If $n$ is a positive even integer, then

$\underset{x\to a}{\lim}\frac{1}{(x-a)^n}=+\infty$ .

If $n$ is a positive odd integer, then

$\underset{x\to a^+}{\lim}\frac{1}{(x-a)^n}=+\infty$

and

$\underset{x\to a^-}{\lim}\frac{1}{(x-a)^n}=−\infty$ .

Nosotros should also indicate out that in the graphs of $f(x)=1/(x-a)^n$ , points on the graph having $x$ -coordinates very most to $a$ are very close to the vertical line $x=a$ . That is, every bit $x$ approaches $a$ , the points on the graph of $f(x)$ are closer to the line $x=a$ . The line $x=a$ is called a vertical asymptote of the graph. We formally ascertain a vertical asymptote as follows:

Definition

Permit $f(x)$ exist a function. If any of the post-obit conditions concord, and then the line $x=a$ is a vertical asymptote of $f(x)$ :

$\begin{array}{ccc}\hfill \underset{x\to a^-}{\lim}f(x)& =\hfill & +\infty \, \text{or} \, -\infty \hfill \\ \hfill \underset{x\to a^+}{\lim}f(x)& =\hfill & +\infty \, \text{or} \, −\infty \hfill \\ & \text{or}\hfill & \\ \hfill \underset{x\to a}{\lim}f(x)& =\hfill & +\infty \, \text{or} \, −\infty \hfill \end{array}$

Finding a Vertical Asymptote

Solution

a. $\underset{x\to 2^-}{\lim}\frac{1}{(x-2)^3}=−\infty$ ;

b. $\underset{x\to 2^+}{\lim}\frac{1}{(x-2)^3}=+\infty$ ;

c. $\underset{x\to 2}{\lim}\frac{1}{(x-2)^3}$ DNE. The line $x=2$ is the vertical asymptote of $f(x)=1/(x-2)^3$ .

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

Behavior of a Function at Dissimilar Points

Chapter Opener: Einstein'south Equation

A picture of a futuristic spaceship speeding through deep space. — Figure eleven. (credit: NASA)

In the chapter opener we mentioned briefly how Albert Einstein showed that a limit exists to how fast any object can travel. Given Einstein's equation for the mass of a moving object, what is the value of this bound?

Solution

Our starting point is Einstein'southward equation for the mass of a moving object,

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$ ,

where $m_0$ is the object's mass at rest, $v$ is its speed, and $c$ is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses $m/m_0$ as a function of the ratio of speeds, $v/c$ ((Figure)).

A graph showing the ratio of masses as a function of the ratio of speed in Einstein's equation for the mass of a moving object. The x axis is the ratio of the speeds, v/c. The y axis is the ratio of the masses, m/m0. The equation of the function is m = m0 / sqrt(1 – v2 / c2 ). The graph is only in quadrant 1. It starts at (0,1) and curves up gently until about 0.8, where it increases seemingly exponentially; there is a vertical asymptote at v/c (or x) = 1. — Figure 12. This graph shows the ratio of masses as a function of the ratio of speeds in Einstein's equation for the mass of a moving object.

Nosotros can run across that as the ratio of speeds approaches 1—that is, every bit the speed of the object approaches the speed of light—the ratio of masses increases without spring. In other words, the function has a vertical asymptote at $v/c=1$ . We tin can endeavor a few values of this ratio to test this idea.

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

Thus, co-ordinate to (Figure), if an object with mass 100 kg is traveling at 0.9999 $c$ , its mass becomes 7071 kg. Since no object can have an infinite mass, we conclude that no object can travel at or more than the speed of calorie-free.

Key Concepts

A table of values or graph may be used to estimate a limit.
If the limit of a function at a signal does not exist, it is still possible that the limits from the left and right at that signal may exist.
If the limits of a function from the left and right be and are equal, then the limit of the function is that common value.
We may use limits to describe infinite behavior of a role at a point.

Key Equations

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

2.What exercise your results in the preceding exercise indicate about the 2-sided limit $\underset{x\to 1}{\lim}f(x)$ ? Explain your response.

Solution

$\underset{x\to 1}{\lim}f(x)$ does non exist because $\underset{x\to 1^-}{\lim}f(x)=-2 \ne \underset{x\to 1^+}{\lim}f(x)=2$ .

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

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

Solution

$\underset{x\to 0}{\lim}(1+x)^{1/x}=2.7183$

5.To which mathematical abiding does the limit in the preceding practice appear to be getting closer?

In the following exercises, use the given values of $x$ to set up a tabular array to evaluate the limits. Round your solutions to eight decimal places.

Solution

a. 1.98669331; b. ane.99986667; c. 1.99999867; d. ane.99999999; e. 1.98669331; f. ane.99986667; m. 1.99999867; h. 1.99999999; $\underset{x\to 0}{\lim}\frac{\sin 2x}{x}=2$

8.Use the preceding two exercises to theorize (guess) the value of the following limit: $\underset{x\to 0}{\lim}\frac{\sin ax}{x}$ for $a$ , a positive real value.

Solution

$\underset{x\to 0}{\lim}\frac{\sin ax}{x}=a$

In the post-obit exercises, set up a table of values to observe the indicated limit. Round to eight digits.

Solution

a. −0.80000000; b. −0.98000000; c. −0.99800000; d. −0.99980000; e. −1.2000000; f. −one.0200000; thousand. −1.0020000; h. −1.0002000;

$\underset{x\to 1}{\lim}(1-2x)=-1$

Solution

a. −37.931934; b. −3377.9264; c. −333,777.93; d. −33,337,778; eastward. −29.032258; f. −3289.0365; g. −332,889.04; h. −33,328,889

$\underset{x\to 0}{\lim}\frac{z-1}{z^2(z+3)}=−\infty$

Solution

a. 0.13495277; b. 0.12594300; c. 0.12509381; d. 0.12500938; east. 0.11614402; f. 0.12406794; 1000. 0.12490631; h. 0.12499063;

$\underset{x\to 2}{\lim}\frac{1-\frac{2}{x}}{x^2-4}=0.1250=\frac{1}{8}$

In the post-obit exercises, set up a table of values and round to eight meaning digits. Based on the tabular array of values, make a guess about what the limit is. Then, utilize a calculator to graph the function and decide the limit. Was the conjecture correct? If not, why does the method of tables neglect?

Solution

a. −10.00000; b. −100.00000; c. −1000.0000; d. −10,000.000; Guess: $\underset{\alpha \to 0^+}{\lim}\frac{1}{\alpha } \cos (\frac{\pi }{\alpha })=\infty$ , 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 office $y=f(x)$ shown hither. Which of the statements about $y=f(x)$ are truthful and which are faux? Explain why a argument is false.

17. $\underset{x\to 10}{\lim}f(x)=0$

xviii. $\underset{x\to -2^+}{\lim}f(x)=3$

Solution

Imitation; $\underset{x\to -2^+}{\lim}f(x)=+\infty$

19. $\underset{x\to -8}{\lim}f(x)=f(-8)$

20. $\underset{x\to 6}{\lim}f(x)=5$

In the following exercises, utilise the following graph of the function $y=f(x)$ to find the values, if possible. Gauge when necessary.

i. The first segment is linear with a slope of i and goes through the origin. Its endpoint is a closed circle at (1,1). The 2d segment is also linear with a slope of -one. It begins with the open circle at (i,2).">

21. $\underset{x\to 1^-}{\lim}f(x)$

22. $\underset{x\to 1^+}{\lim}f(x)$

23. $\underset{x\to 1}{\lim}f(x)$

24. $\underset{x\to 2}{\lim}f(x)$

25. $f(1)$

In the following exercises, use the graph of the office $y=f(x)$ shown here to find the values, if possible. Estimate when necessary.

A graph of a piecewise function with two segments. The beginning 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).

26. $\underset{x\to 0^-}{\lim}f(x)$

27. $\underset{x\to 0^+}{\lim}f(x)$

28. $\underset{x\to 0}{\lim}f(x)$

29. $\underset{x\to 2}{\lim}f(x)$

In the following exercises, utilise the graph of the function $y=f(x)$ shown here to notice the values, if possible. Estimate when necessary.

2, has a gradient of 1, and begins at the open circle (two,2).">

30. $\underset{x\to -2^-}{\lim}f(x)$

31. $\underset{x\to -2^+}{\lim}f(x)$

32. $\underset{x\to -2}{\lim}f(x)$

33. $\underset{x\to 2^-}{\lim}f(x)$

34. $\underset{x\to 2^+}{\lim}f(x)$

35. $\underset{x\to 2}{\lim}f(x)$

In the following exercises, apply the graph of the function $y=g(x)$ shown hither to observe the values, if possible. Estimate when necessary.

=0 and is the left half of an upward opening parabola with vertex at the airtight circle (0,iii). The second exists for ten>0 and is the correct half of a downward opening parabola with vertex at the open circumvolve (0,0).">

36. $\underset{x\to 0^-}{\lim}g(x)$

37. $\underset{x\to 0^+}{\lim}g(x)$

38. $\underset{x\to 0}{\lim}g(x)$

In the following exercises, apply the graph of the function $y=h(x)$ shown here to find the values, if possible. Approximate 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.

39. $\underset{x\to 0^-}{\lim}h(x)$

40. $\underset{x\to 0^+}{\lim}h(x)$

41. $\underset{x\to 0}{\lim}h(x)$

In the post-obit exercises, use the graph of the function $y=f(x)$ shown here to detect the values, if possible. Gauge when necessary.

0, and there is a closed circle at the origin.">

42. $\underset{x\to 0^-}{\lim}f(x)$

43. $\underset{x\to 0^+}{\lim}f(x)$

44. $\underset{x\to 0}{\lim}f(x)$

45. $\underset{x\to 1}{\lim}f(x)$

46. $\underset{x\to 2}{\lim}f(x)$

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

47. $\underset{x\to 2}{\lim}f(x)=1, \, \underset{x\to 4^-}{\lim}f(x)=3, \, \underset{x\to 4^+}{\lim}f(x)=6$ , the function is not defined at $x=4$ .

48. $\underset{x\to -\infty }{\lim}f(x)=0, \, \underset{x\to -1^-}{\lim}f(x)=−\infty$ , $\underset{x\to -1^+}{\lim}f(x)=\infty, \, \underset{x\to 0}{\lim}f(x)=f(0), \, f(0)=1, \, \underset{x\to \infty }{\lim}f(x)=−\infty$

Solution

Answers may vary.

49. $\underset{x\to -\infty}{\lim}f(x)=2, \, \underset{x\to 3^-}{\lim}f(x)=−\infty$ , $\underset{x\to 3^+}{\lim}f(x)=\infty, \, \underset{x\to \infty }{\lim}f(x)=2, \, f(0)=\frac{-1}{3}$

50. $\underset{x\to -\infty }{\lim}f(x)=2, \, \underset{x\to -2}{\lim}f(x)=−\infty$ , $\underset{x\to \infty }{\lim}f(x)=2, \, f(0)=0$

Solution

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.

51. $\underset{x\to -\infty }{\lim}f(x)=0, \, \underset{x\to -1^-}{\lim}f(x)=\infty, \, \underset{x\to -1^+}{\lim}f(x)=−\infty, \, f(0)=-1, \, \underset{x\to 1^-}{\lim}f(x)=−\infty, \, \underset{x\to 1^+}{\lim}f(x)=\infty, \, \underset{x\to \infty }{\lim}f(x)=0$

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Source: https://opentextbc.ca/calculusv1openstax/chapter/the-limit-of-a-function/

Refer Again to the Graph Figure 1 Select All True Statements About the Graph

2.ii The Limit of a Function

Learning Objectives

Intuitive Definition of a Limit

Problem-Solving Strategy: Evaluating a Limit Using a Table of Functional Values

Evaluating a Limit Using a Table of Functional Values 1

Evaluating a Limit Using a Table of Functional Values 2

Solution

Evaluating a Limit Using a Graph

Solution

2 Of import Limits

The Existence of a Limit

Evaluating a Limit That Fails to Exist

Solution

One-Sided Limits

Evaluating One-Sided Limits

Solution

Space Limits

Recognizing an Space Limit

Infinite Limits from Positive Integers

Definition

Finding a Vertical Asymptote

Solution

Behavior of a Function at Dissimilar Points

Chapter Opener: Einstein'south Equation

Solution

Key Concepts

Key Equations

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

Solution

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