# Limit

# Limit

In mathematics the concept of limit formally expresses the notion of arbitrary closeness. That is, a limit is a value that a variable quantity approaches as closely as one desires. The operations of differentiation and integration from calculus are both based on the theory of limits.

The theory of limits is based on a particular property of the real numbers; namely that between any two real numbers, no matter how close together they are, there is always another one. Between any two real numbers there are always infinitely many more.

Nearness is key to understanding limits: only after nearness is defined does a limit acquire an exact meaning. Relevantly, a neighborhood of points near any given point comprise a neighborhood. Neighborhoods are definitive components of infinite limits of a sequence.

## History

Ancient Greek mathematician Archimedes of Syracuse (287–212 BC) first developed the idea of limits to measure curved figures and the volume of a sphere in the third century BC. By carving these figures into small pieces that can be approximated, then increasing the number of pieces, the limit of the sum of pieces can give the desired quantity. Archimedes’ thesis,*The Method,* was lost until 1906, when mathematicians discovered that Archimedes came close to discovering infinitesimal calculus.

As Archimedes’ work was unknown until the twentieth century, others developed the modern mathematical concept of limits. English physicist and mathematician Sir Isaac Newton (1642–1727) and German mathematician Gottfried Wilhelm Leibniz (1646–1716) independently developed the general principles of calculus (of which the theory of limits is an important part) in the seventeenth century.

## Limit of a sequence

Ancient Greek philosopher (of southern Italy) Zeno of Elea (c 490–c 430 BC) may have been one of the first mathematicians to ponder the limit of a sequence and wonder how it related to the world around him. Zeno argued that all motion was impossible because in order to move a distance (l) it is first necessary to travel half the distance, then half the remaining distance, then half of that remaining distance and so on. Thus, he argued, the distance (l) can never be fully traversed.

Consider the sequence 1, 1/2, 1/4, 1/8,. . .(1/2)^{n} when n gets very large. Since (1/2)^{n} equals 1/2 multiplied by itself n times, (1/2)^{n} gets very small when n is allowed to become infinitely large. The sequence is said to converge, meaning numbers that are very far along in the sequence (corresponding to large “N”) get very close together and very close to a single value called the limit.

A sequence of numbers converges to a given number if the differences between the terms of the sequence and the given number form an infinitesimal sequence. For this sequence (1/2)^{n} gets arbitrarily close to 0, so 0 is the limit of the sequence. The numbers in the sequence never quite reach the limit, but they never go past it either.

If an infinite sequence diverges, the running total of the terms eventually turns away from any specific value, so a divergent sequence has no limiting sum.

## Limit of a function

Consider an arbitrary function, y = *f* (x). (A function is a set of ordered pairs for which the first and second elements of each pair are related to one another in a fixed way. When the elements of the ordered pairs are real numbers, the relationship is usually expressed in the form of an equation.) Suppose that successive values of x are chosen to match those of a converging sequence such as the sequence S from the previous example. The question arises as to what the values of the function do, that is, what happens to successive values of y. In fact, whenever the values of x form a sequence, the values *f* (x) also form a sequence. If this sequence is a converging sequence then the limit of that sequence is called the limit of the function. More generally when the value of a function *f* (x) approaches a definite value L as the independent variable x gets close to a real number p then L is called the limit of the function. This is written formally as:

lim f(x) = l

x → p

and reads “The limit of f of x, as x approaches p, equals L.” It does not depend on what particular sequence of numbers is chosen to represent x; it is only necessary that the sequence converge to a limit. The limit may depend on whether the sequence is increasing or decreasing. That is the limit, as x approaches p from above may be different from the limit as x approaches p from below. In some cases, one or the other of these limits may even fail to exist. In any case since the value of x is approaching the finite value p the difference (p x) is approaching zero. It is this definition of limit that provides a foundation for development of the derivative and the integral in calculus.

There is a second type of functional limit: the limit as the value of the independent variable approaches infinity. While a sequence that approaches infinity is said to diverge, there are cases for which applying the defining rule of a function to a diverging sequence results in creation of a converging sequence. The function defined by the equation y = 1/x is such a function. If a finite limit exists for the function when the

### KEY TERMS

**Converge** —To converge is to approach a limit that has a finite value.

**Interval** —An interval is a subset of the real numbers corresponding to a line segment of finite length, and including all the real numbers between its end points. An interval is closed if the endpoints are included and open if they are not.

**Real Number** —The set of numbers containing the integers and all the decimals including both the repeating and nonrepeating decimals.

**Sequence** —A sequence is a series of terms, in which each successive term is related to the one before it by a fixed formula.

independent variable approaches infinity it is written formally as:

lim f (x) = L

x → ∞

and reads “The limit of f of x, as x approaches infinity, equals L.” It is interesting to note that the function defined by y = 1/x has no limit when x approaches 0 but has the limit L = 0 when x approaches ∞.

## Applications

The limit concept is essential to understanding the real number system and its distinguishing characteristics. In one sense real numbers can be defined as the numbers that are the limits of convergent sequences of rational numbers. One application of the concept of limits is on the derivative. The derivative is a rate of flow or change, and can be computed based on some limits concepts. Limits are also key to calculating integrals (expressions of areas). The integral calculates the entire area of a region by summing up an infinite number of small pieces of it. Limits are also part of the iterative process. An iteration is repeatedly performing a routine, using the output of one step as the input of the next step. Each output is an iterate. Some successful iterates can get as close as desired to a theoretically exact value.

## Resources

### BOOKS

Abbott, Percival. *Teach Yourself: Calculus.* London, UK: Hodder and Stoughton Education, and Chicago, IL: Contemporary Books, 2003.

Burton, David M. *The History of Mathematics: An Introduction.* New York: McGraw-Hill, 2007.

Larson, Ron. *Calculus: An Applied Approach.* Boston, MA: Houghton Mifflin, 2003.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Lyublinskava, Irina E. *Connecting Mathematics with Science: Experiments for Precalculus.* Emeryville, CA: Key Curriculum Press, 2003.

Setek, William M. *Fundamentals of Mathematics.* Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

J. R. Maddocks

# Limit

# Limit

In **mathematics** the concept of limit formally expresses the notion of arbitrary closeness. That is, a limit is a value that a **variable** quantity approaches as closely as one desires. The operations of differentiation and integration from **calculus** are both based on the theory of limits. The theory of limits is based on a particular property of the **real numbers** ; namely that between any two real numbers, no matter how close together they are, there is always another one. Between any two real numbers there are always infinitely many more.

Nearness is key to understanding limits: only after nearness is defined does a limit acquire an exact meaning. Relevantly, a neighborhood of points near any given point comprise a neighborhood. Neighborhoods are definitive components of infinite limits of a sequence.

## History

Archimedes of Syracuse first developed the idea of limits to measure curved figures and the **volume** of a **sphere** in the third century b.c. By carving these figures into small pieces that can be approximated, then increasing the number of pieces, the limit of the sum of pieces can give the desired quantity. Archimedes' thesis, *The Method*, was lost until 1906, when mathematicians discovered that Archimedes came close to discovering infinitesimal calculus.

As Archimedes' work was unknown until the twentieth century, others developed the modern mathematical concept of limits. Englishman Sir Issac Newton and German Gottfried Wilhelm von Leibniz independently developed the general principles of calculus (of which the theory of limits is an important part) in the seventeenth century.

## Limit of a sequence

The ancient Greek philosopher Zeno may have been one of the first mathematicians to ponder the limit of a sequence and wonder how it related to the world around him. Zeno argued that all **motion** was impossible because in order to move a **distance** l it is first necessary to travel half the distance, then half the remaining distance, then half of that remaining distance and so on. Thus, he argued, the distance l can never be fully traversed.

Consider the sequence 1, 1/2, 1/4, 1/8,...(1/2)n when n gets very large. Since (1/2)n equals 1/2 multiplied by itself n times, (1/2)n gets very small when n is allowed to become infinitely large. The sequence is said to converge, meaning numbers that are very far along in the sequence (corresponding to large n) get very close together and very close to a single value called the limit.

A sequence of numbers converges to a given number if the differences between the terms of the sequence and the given number form an infinitesimal sequence. For this sequence (1/2)n gets arbitrarily close to 0, so 0 is the limit of the sequence. The numbers in the sequence never quite reach the limit, but they never go past it either.

If an infinite sequence diverges, the running total of the terms eventually turns away from any specific value, so a divergent sequence has no limiting sum.

## Limit of a function

Consider an arbitrary **function** , y = f(x). (A function is a set of ordered pairs for which the first and second elements of each pair are related to one another in a fixed way. When the elements of the ordered pairs are real numbers, the relationship is usually expressed in the form of an equation.) Suppose that successive values of x are chosen to match those of a converging sequence such as the sequence S from the previous example. The question arises as to what the values of the function do, that is, what happens to successive values of y. In fact, whenever the values of x form a sequence, the values f(x) also form a sequence. If this sequence is a converging sequence then the limit of that sequence is called the limit of the function. More generally when the value of a function f(x) approaches a definite value L as the independent variable x gets close to a real number p then L is called the limit of the function. This is written formally as:

and reads "The limit of f of x, as x approaches p, equals L." It does not depend on what particular sequence of numbers is chosen to represent x; it is only necessary that the sequence converge to a limit. The limit may depend on whether the sequence is increasing or decreasing. That is the limit, as x approaches p from above may be different from the limit as x approaches p from below. In some cases one or the other of these limits may even fail to exist. In any case since the value of x is approaching the finite value p the difference (p-x) is approaching **zero** . It is this definition of limit that provides a foundation for development of the **derivative** and the **integral** in calculus.

There is a second type of functional limit: the limit as the value of the independent variable approaches **infinity** . While a sequence that approaches infinity is said to diverge, there are cases for which applying the defining rule of a function to a diverging sequence results in creation of a converging sequence. The function defined by the equation y = 1/x is such a function. If a finite limit exists for the function when the independent variable approaches infinity it is written formally as:

and reads "The limit of f of x, as x approaches infinity, equals L." It is interesting to note that the function defined by y = 1/x has no limit when x approaches 0 but has the limit L = 0 when x approaches ∞.

## Applications

The limit concept is essential to understanding the real number system and its distinguishing characteristics. In one sense real numbers can be defined as the numbers that are the limits of convergent **sequences** of rational numbers. One application of the concept of limits is on the derivative. The derivative is a **rate** of flow or change, and can be computed based on some limits concepts. Limits are also key to calculating intergrals (expressions of areas). The integral calculates the entire area of a region by summing up an infinite number of small pieces of it. Limits are also part of the iterative process. An **iteration** is repeatedly performing a routine, using the output of one step as the input of the next step. Each output is an iterate. Some successful iterates can get as close as desired to a theoretically exact value.

## Resources

### books

Abbot, P., and M.E. Wardle. *Teach Yourself Calculus.* Lincolnwood: NTC Publishing, 1992.

Allen, G.D., C. Chui, and B. Perry. *Elements of Calculus.* 2nd ed. Pacific Grove: Brooks/Cole Publishing Co., 1989.

Gowar, Norman. *An Invitation to Mathematics.* New York: Oxford University Press, 1979.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Silverman, Richard A. *Essential Calculus With Applications.* New York: Dover, 1989.

### periodicals

McLaughlin, William I. "Resolving Zeno's Paradoxes." *Scientific American* 271 (1994): 84-89.

J. R. Maddocks

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Converge**—To converge is to approach a limit that has a finite value.

**Interval**—An interval is a subset of the real numbers corresponding to a line segment of finite length, and including all the real numbers between its end points. An interval is closed if the endpoints are included and open if they are not.

**Real Number**—The set of numbers containing the integers and all the decimals including both the repeating and nonrepeating decimals.

**Sequence**—A sequence is a series of terms, in which each successive term is related to the one before it by a fixed formula.

# Limit

# Limit

The concept of limit is an essential component of **calculus** . Limits are typically the first idea of calculus that students study. Two fundamental concepts in calculus—the **derivative** and the **integral** —are based on the limit concept. Limits can be examined using three intuitive approaches: number sequences, functions, and geometric shapes.

## Number Sequences

One way to examine limits is through a sequence of numbers. The following example shows a sequence of numbers in which the limit is 0.

The second number in the sequence, ½, is the result of dividing the first number in the sequence, 1, by 2. The third number in the sequence, ¼, is the result of dividing the second number in the sequence, ½, by 2.

This process of dividing each number by 2 to acquire the next number in the sequence is continued in order to acquire each of the remaining values. The three dots indicate that the sequence does not end with the last number that appears in the list, but rather that the sequence continues infinitely.

If the sequence continues infinitely, the values in the sequence will get closer and closer to 0. The numbers in the sequence, however, will never actually take on the value of zero. The mathematical concept of approaching a value without reaching that value is referred to as the "limit concept." The value that is being approached is called the limit of the sequence. The limit of the sequence … is 0.

The example below displays several sequences and their limits. In each case, the values in the sequence are getting closer to their limit.

Example 1: 0.9, 0.99, 0.999, 0.9999, 0.99999, 0.999999, 0.9999999, … Limit: 1

Example 2: 5.841, 5.8401, 5.84001, 5.840001, 5.8400001, 5.84000001, … Limit: 5.84

Example 3: … Limit: 0

Not all sequences, however, have limits. The sequence 1, 2, 3, 4… increases and does not approach a single value. Another example of a sequence that has no limit is -1.1, 2.2, -3.3, 4.4, -5.5, 6.6, …. Because there is no specific number that this sequence approaches, the sequence has no limit.

## Functions

Limits can also be examined using functions. An example of a function is . One way to examine the limit of a function is to list a sample of the values that comprise the function. The left-hand portion of the table can be used to examine the limit of the function as *x* increases.

As the values in the *x* column increase, the values in the *f* (*x* ) column get closer to 0. The limit of a function is equal to the value that the *f* (*x* ) column approaches. The limit of the function as *x* approaches infinity is 0.

Functions can also be plotted on a **Cartesian plane** . A graph of the function is shown in the figure. The color curve represents the function. As the *x* values increase, the color curve or the *f* (*x* ) values get closer and closer to 0. Once again, the limit of the function as *x* goes to infinity is 0.

It is important to consider what value *x* is approaching when determining the limit of *f* (*x* ). If *x* were approaching 0 in the preceding example, *f* (*x* ) would not have a limit. The reason for this can be understood using the middle and right-hand portions of the table.

The table suggests that the values for *f* (*x* ) continue to increase as *x* approaches 0 from values that are greater than 0. The table also suggests that the values for *f* (*x* ) continue to decrease as *x* approaches 0 from values that are less than 0. Because the *f* (*x* ) values do not approach a specific value, the function does not have a limit as *x* approaches 0.

## Geometric Shapes

A typical application of the limit concept is in finding area. For example, one method for estimating the area of a circle is to divide the circle into small triangles, as shown below, and summing the area of these triangles. The circle in (a) is divided into six triangles. If a better estimate of area is desired, the circle can be divided into smaller triangles as shown in (b).

If the exact area of the circle is needed, the number of triangles that divide the circle can be increased. The limit of sum of the area of these triangles, as the number of triangles approaches infinity, is equal to the standard formula for finding the area of a circle, *A = πr* ^{2}, where *A* is the area of the circle and *r* is its radius.

In summary, limit refers to a mathematical concept in which numerical values get closer and closer to a given value or approaches that value. The value that is being approached is called the "limit." Limits can be used to understand the behavior of number sequences and functions. They can also be used to determine the area of geometric shapes. By extending the process that is used for finding the area of a geometric shape, the volume of geometric solids can also be found using the limit concept.

see also Calculus; Infinity.

*Barbara M. Moskal*

## Bibliography

Jockusch, Elizabeth A., and Patrick J. McLoughlin. "Implementing the Standards: Building Key Concepts for Calculus in Grades 7–12." *Mathematics Teacher* 83, no. 7 (1990): 532–540.

### Internet Resources

"Limits" Coolmath.com. <http://www.coolmath.com/limit1.htm>.

# limit

lim·it / ˈlimit/ •
n. 1. a point or level beyond which something does not or may not extend or pass: *the limits of presidential power*

*the 10-minute*

**limit on**speeches*there was no*∎ (often limits) the terminal point or boundary of an area or movement:

**limit to**his imagination.*the city limits*

*the upper limit of the tidal reaches.*∎ the furthest extent of one's physical or mental endurance:

*Mary Ann tried everyone's patience*

**to the limit***other horses were reaching their limit.*2. a restriction on the size or amount of something permissible or possible:

*an age limit*

*a weight limit.*∎ a speed limit:

*a 30 mph limit.*∎ (in card games) an agreed maximum stake or bet. ∎ (also legal limit) the maximum concentration of alcohol in the blood that the law allows in the driver of a motor vehicle:

*the risk of drinkers inadvertently going*3. Math. a point or value that a sequence, function, or sum of a series can be made to approach progressively, until it is as close to the point or value as desired.• v. (lim·it·ed, lim·it·ing) [tr.] set or serve as a limit to:

**over the limit**.*try to limit the amount you drink*

*class sizes are*[as adj.] (limiting)

**limited to**a maximum of 10*a limiting factor.*PHRASES: be the limit inf. be intolerably troublesome or irritating.off limits out of bounds:

*they declared the site off limits*fig.

*there was no topic that was off limits for discussion.*within limits moderately; up to a point:without limit with no restriction.DERIVATIVES: lim·i·ta·tive / ˈliməˌtātiv/ adj.

# Limitation

# LIMITATION

The term limitation is used in philosophy and theology to explain why some being or some property of a being is only limited or finite, as in the case of creatures, and not unlimited or infinite, as in the case of God. A principal problem in scholastic philosophy is to explain, in terms of causes or principles, the limitation of one creature compared to another (e.g., man to angels), and more especially of all creatures compared to God.

Thomists usually assign a twofold reason for the qualitative limitation of a given being. First, every limited being requires some external agent or efficient cause to determine its capacity or limit, and to communicate the corresponding degree of perfection. Second, the result or effect in the being of the determining action of its cause is some internal principle of limitation within the being itself. This inner principle of limitation fixes the being's inner capacity for receiving so much and no more of a given attribute or perfection. To borrow an analogy from the quantitative order, if a person wishes to fill a pitcher with water, he must pour so much water; likewise the pitcher (the recipient) must itself have a certain shape or capacity to be able to receive the water. This inner cause or principle of limitation St. thomas aquinas called a potency for receiving a perfection or act (see potency and act). Both of these terms he borrowed from aristotle, the original proponent of potency and act, though Aristotle himself applied his theory only to the problem of change and not to that of limitation, St. Thomas argues that no positive qualitative perfection, such as knowledge, goodness, or power, can be identically its own limiting principle, i.e., the reason why it is possessed by a particular being to a limited degree and not in its fullness. Therefore, wherever there is limitation there must be an internal duality or real metaphysical distinction of elements within the limited being: one principle to take care of the positive perfection that is received or participated; the other to limit the capacity of the subject that receives or participates. This philosophical doctrine, referred to as the limitation of act by potency, may be summarized thus: No act (or perfection) can be found in a limited state unless it be received into a really distinct limiting potency.

Other scholastic philosophers, such as John duns scotus and F. suÁrez, agree with St. Thomas on the need for an external agent to determine the limitation of a being, but deny that any internal principle of potency within the limited being need be really distinct from the perfection it limits.

See Also:finite being; participation; perfection, ontological.

**Bibliography:** w. n. clarke, "Limitation of Act by Potency," *The New Scholasticism* 26 (1952) 167–194. g. giannini, *Enciclopedia filosofica,* 4 v. (Venice–Rome 1957) 3:54–58.

[w. n. clarke]

# limitation

lim·i·ta·tion / ˌliməˈtāshən/ •
n. 1. (often limitations) a limiting rule or circumstance; a restriction: *severe limitations on water use.* ∎ a condition of limited ability; a defect or failing:

*she knew her limitations better than she knew her worth.*∎ the action of limiting something:

*the limitation of local authorities' powers.*2. (also limitation period) Law a legally specified period beyond which an action may be defeated or a property right is not to continue.See also statute of limitations.

# Limitation

# LIMITATION

*A qualification, restriction, or circumspection.*

In the law of property, a limitation on an estate arises when its duration or quality is in some way restricted. For example, in the conveyance, "Owner conveys Blackacre to A until B leaves the country," A's estate is limited, since A is given Blackacre for only a specified length of time.

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