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 = πr2, 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.
