Limit of a Function and Limit Laws (Part I)

Limit of a Function and Limit Laws (Part I)

Limit of a function is the maximum value that the graph of a function can take. The limit of a function is equal to the value of the function at the point where the graph intersects the x-axis on the left hand side. If we draw the graph of y = f(x), then its limit at x = a is equal to f(a).

A zero denominator is a fraction whose numerator and denominator are both 0.

For example, the fraction 1/0 = undefined because there are no numbers that can multiply together to produce 0. Solving for a zero denominator means canceling the common factor, if possible, or simplifying the expression until it becomes a number.

The Sandwich Theorem states that the sum of the areas of two non-overlapping regions is equal to the area of the region formed by their union. In other words, if you take two shapes and place them together, you can create a third shape where the combined area of both shapes equals the area of their new shape.

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We will consider more formally the concept of limit of a function and some laws which allow us to compute limits of more complicated functions.

Concept of the limit of a function at x – intuitive approach.

Techniques to evaluate the limit:

• Limit laws
• Eliminate zero denominator
• The sandwich Theorem

To understand the concept of the limit of a function, we can consider an example of a falling rock. This motivates the intuitive approach to this concept, which is to try to look at the behavior of the function when an h tends to zero or more generally when we try to look at a nearby point that approaches that particular point.

After that, we will consider some techniques for evaluating limits involving zero denominators. Then we will try to eliminate the zero denominator by using limit laws and finally we will look at a powerful theorem called Sandwich Theorem.

As we have seen in the example of the falling rock, we try to study a function's behavior around a point of interest.

The limit of a function f(x) as x approaches a particular value Xo exists if the values of f(x) get close to this particular value L as x gets close to Xo.

In this case, we write the limit of f(x) as L equals to the limit of f(x) when x tends to Xo. And be careful.

In this notation, the value of F(Xo) is not considered, but rather the value of f(x) when x is near to Xo but not exactly equal to Xo.

Let's look at the following graph, which illustrates the situation. The blue line is a graph of the function y = f(x). This point (also known as Xo) is where we want to focus our attention, and observe how the value of the function behaves as x approaches it.

Thus the value of the function f approaches this point L as x tends to Xo.

Here are some simple examples for you.

This first example is given by f(x) = x² − 1. The numerator is x² − 1, and the denominator is x − 1.

Now, if we draw the graph of the function, we see that it is a straight line except at the point where x equals 1, where there is a hole. This means that the function has no meaning when x equals 1 because if you put in x equals 1, both the numerator and

denominator equal 1 and are therefore meaningless. So actually this function is really this straight line except with a hole here which is not defined when x equals 1.

If we look at this graph here, we see that f(x) equals x + 1 is a straight line, very much like the one on the left except that it does not have a hole because even at x = 1, it has a value of 2. It is defined for x equals to 1 as well.

One must differentiate between the two functions, as they differ at one single point. The limit of f(x) as x tends to one depends on whether that single point is within the domain of f(x) or not. We need to find the limit.

Although the graphs are not identical, you can see that as x approaches one, the function behaves similarly at that point; therefore, it does not matter where x is when it equals one.

In both examples, the limit of f(x) as x approaches 1 is 2. Of course, one must be careful to note that the limit of a function at a point may or may not exist.

Another example of a function is constant function, which means that no matter what value x takes, the value of the function is always the same. That value is called the constant of the function. So graphically, it represents a horizontal line like this.

And now, in this case, you can see that the points Xo here move around the point x. The behavior of the function f(x) is always at this constant value which is represented by the height of this blue line. That is why f(x) is always equal to c, and its limit as x tends to zero is equal to c.

This example shows an identity function whose graph is the blue straight line at 45° with the x-axis.

Now, the function f(x) equals x. This means that the value of this function is just the value of x, which is a 45° line. What is the limit when x tends to zero of this function?

As x approaches Xo, the value of the function approaches Xo. This is because f(x) = x. Thus, for this identity function, lim x tends to Xo is equal to Xo.

The example here has two parts, as can be seen in the graph.

One part of a function is equal to two at the level one, but less than one. Another part is equal to minus two at level one and beyond.

If we consider the point x equals 1, when x gets close to 1 it is on the right-hand side very close to -2 and when it gets close to 2 it is on the left-hand side very close to 2.

When x is close to one, the function does not come close to a particular fixed value. Therefore, the limit of this function as x tends to one does not exist.

The value of the function at one is -2, but the behavior of the function does not approach a fixed value as x approaches one.

In some cases, limits do not exist. For instance, if we approach point one from either side, the limit does not exist.

For example, the graph illustrated here shows two blue lines. If we look at the point x equals zero, the behavior of the function is very different when we approach that point from the right-hand side (which goes to infinity).

If we approach the point x equals zero from the left-hand side, the function goes to minus infinity. Since it does not come close to a particular value, the limit does not exist. The formula for this function is given by f(x) equals one over x if x is nonzero and equal to zero if x equals zero.

As was indicated, the limit of the function does not exist because there are two different values for it when we approach x = 0 from different sides.

The following example is a famous one in calculus. The function f(x) is equal to zero if x is less than or equal to zero.

The function on the left-hand side is a horizontal straight line, which is parallel to the negative x-axis. The function on the right-hand side oscillates between minus one and one as x approaches zero, because as x becomes smaller and smaller, one over x becomes larger and larger. Since sine is periodic with period 2pi, it oscillates between these values: -1 and 1. This causes this oscillation in our function.

If we look at the point x = 0, we see that the behavior of f(x) on both sides is very different. On the left-hand side it is constantly equal to zero while on the right-hand side it oscillates and does not come close to a fixed value. Therefore, since the function does not approach

a fixed value as x approaches zero, the limit does not exist because of the behavior of f(x) on the right-hand side that it oscillates between negative one and positive one.