The concept of the limit in calculus is very important.  It describes what happens to a function as a particular value is approached.  The derivative, one of the major concepts in calculus, is defined in limit terms.  This short section will help you to think in terms of limits.  The first thing to understand about limits is that the limit of a function is not the value of the function. This change in thinking (from value to limit) is important because most functions are understood as a series of mathematical operations that can be evaluated at certain points simply by substitution. This is not true when taking limits.

The polynomial function

can be evaluated for any real number: replace x with the number and perform the indicated operations. Asking for the limit of the function as x approaches two is an uninteresting question, because the function can be evaluated at 2.

Other functions such as polynomial fractions cannot be evaluated at certain points and these functions are best understood by thinking in terms of limits. The function

can be evaluated for any real number except −2. Replacing x by −2 produces the statement 0/0. Recall that any number times 0 is 0, but any number divided by 0 is undefined, including 0/0. The only way to understand how the function behaves in the vicinity of −2 is to look at the limit as x approaches −2. The limit of a function answers the question: "What happens to the function as the limiting value is approached?" The mathematical notation for this operation is

The notation in front of the functions is read "the limit, as x approaches minus two."  In the case of rational functions, factoring and reducing the fraction helps in finding the limit. 

Finding the limit of this function as x→−2 helps in understanding the function. Since the original function gives the meaningless 0/0 at the point where x→−2, the function cannot exist at x = −2. Graphing the function illustrates this point. The simplified function y = x − 2 is a straight line of slope 1 and intercept −2. The function is also a straight line of slope 1 and intercept −2, but it does not exist at x = − 2. This non-existence at this point is illustrated on the graph as an open circle.

LC01 Limit of a fraction

Another category of functions that are understood with the help of limits is polynomial fractions where the higher power polynomial is in the denominator rather than the numerator. The simplest one to look at is y = 1/x. The product of two variables equal to a constant describes certain relationships .  For example pressure and volume for a fixed amount of gas at a fixed temperature is described by pV = a constant.

The relationship xy = 1 or y = 1/x can be graphed, but the concept of limits more easily describes the behavior of the function. Look at the accompanying figure while going through this discussion. The point x = 1, y = 1 is easy as is x = −1, y = −1. The curve goes through these points, but for the moment concentrate on the positive values of x and y. As x is made larger and larger, y becomes smaller and smaller, but it never goes below 0. Likewise as x is made smaller and smaller (much less than 0). y becomes larger and larger. This is expressed with mathematical symbolism as:  As x→+¥, y→0+. Similarly:  As x→0+, y→+¥ If x is a very small number, then 1 divided by this small number is a very large number. So as x→0+, 1/0+→+¥. With this information the positive region of the graph can be drawn. Some problems dictate only the positive region is physically possible. The graph, however, shows both the positive and the negative region.

LC02 Graphing y = 1/x − 2

LC03 Graphing y = 1/x²

LC04 Graphing Algebraic Fractions

LC05 Limits as x→¥

LC06 Another limit as x→∞

This problem illustrates a manipulative rule for limits.

Similarly

Continuous functions are defined over specific intervals. Continuous functions 1) exist at every point in the defined interval, and 2) have limits at every point equal to the value of the function at that point. Operationally, continuous functions are ones you can draw without lifting your pen.

A discontinuous function is one that either 1) does not exist at some point, or 2) has limits from the positive and negative side at some point that are not equal, ie. the function abruptly changes slope. The firs four problems in this section are examples of discontinuous functions.

Another often used sample of a discontinuous function is the integer function, y = [x], where the [x] notation is understood as meaning "the largest integer contained in x." For example, the lagest integer contained in 3 is 3, and the largest integer contained in 9.9 is 9. Notice that the limit approached from the positive side is different from the limit approached from the negative side. The discontinuous nature of this function is shown in the accompanying figure. In the figure the closed circles indicate the function is defined at the point while the open circles indicate the function is not defined at the point.

Another example of a discontinuous function is one defined on certain intervals such as This is a discontinuous function. Though it is defined everywhere over the interval, the limit as zero is approached from the positive side is 4, the the limit as zero is approached from the negative side is 3. Also note that the function is not defined at zero.

Another example of a discontinuous function is the square root function. The function square root of x is not defined for negative numbers because there are no real roots of negative numbers. In mathematical symbolism

The square root function is continuous from the right side but ends at zero.

In contrast the cube root function behaves differently. There are positive and negative real cube roots of both positive and negative functions so the cube root function is continuous over all values.