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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 calculusthe 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.

Barbara M. Moskal

Bibliography

Jockusch, Elizabeth A., and Patrick J. McLoughlin. "Implementing the Standards: Building Key Concepts for Calculus in Grades 712." Mathematics Teacher 83, no. 7 (1990): 532540.

Internet Resources

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

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limit

limit, in mathematics, value approached by a sequence or a function as the index or independent variable approaches some value, possibly infinity. For example, the terms of the sequence 1/2, 1/4, 1/8, 1/16, … are obviously getting smaller and smaller; since, if enough terms are taken, one can make the last term as small, i.e., as close to zero, as one pleases, the limit of this sequence is said to be zero. Similarly, the sequence 3, 5, 31/2, 41/2, 33/4, 41/4, 37/8, 41/8, … is seen to approach 4 as a limit. However, the sequences 1, 2, 4, 8, 16, … and 1, 2, 1, 2, 1, 2, … do not have limits. Frequently a sequence is denoted by giving an expression for the nth term, sn; e.g., the first example is denoted by sn = 1/2n. The limit, s, of a sequence can then be expressed as lim sn = s, or in the case of the example, lim 1/2n = 0 (read "the limit of 1/2n as n approaches infinity is zero" ). A sequence is a special case of a function. In many functions commonly encountered, the values of the independent variable (the domain) and those of the dependent variable (the range) may be any numbers, while for a sequence the domain is restricted to the positive integers, 1, 2, 3, … . The function y = 1/2x resembles the sequence used as an example, but note that x can take on values other than 1, 2, 3, … ; thus we find not only lim 1/2x = 0 but also lim 1/2x = 4. A more precise definition of the limit of a function is: The function y = f(x) approaches a limit L as x approaches some number a if, for any positive number ε, there is a positive number δ such that |f(x) - L| < ε if 0 < |x - a| < δ. Similarly, f(x) has the limit L as x becomes infinite if for any positive ε there is a δ such that |f(x) - L| < ε if |x| > δ.

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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 limit to his imagination. ∎  (often limits) the terminal point or boundary of an area or movement: 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 over the limit. 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: try to limit the amount you drink class sizes are limited to a maximum of 10 [as adj.] (limiting) 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.

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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.

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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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limit

limit sb. XIV. L. līmes, līmit— frontier.
So limit vb. XIV. — (O)F. limiter or L. līmitāre. limitation XIV.

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limit

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