Limits of the Extreme
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Introduction

    This lesson page will how to calculate the limit of a function for extreme values. Here are the sections within this lesson page:

    Before continuing with this lesson, it is important that you understand what functions are. Use these lessons to learn about functions before proceeding.

    esson: Functions
    esson: Piecewise Functions


    When we examine functions, we look at specific values and evaluate those functions [see our lesson: Evaluating Functions] at those values. While looking at the function f(x) = x + 1, we can evaluate it at x = 4, for instance. We write that evaluation as f(4) = 4 + 1 = 5. The x-value is 4 and the corresponding y-value is 5.

    We can also examine the graph of this function, which is a line.

Linear Graph y=x+1

    The graph can be used to quickly evaluate the functions at various x-values. The graph can also be used to see a general trend or pattern.


    Limits behave differently compared to the evaluation process of functions. Instead of finding y-values for specific x-values, we look to see where the trend for y-values approaches as we get closer and closer to certain x-values.

    The paragraph above can best be understood when we examine the graphs of functions. Let us look at the graph of g(x) = x2, drawn below.

Quadratic Graph y=x^2

    If we wanted to know what is happening to the y-value as the x-values are getting bigger and bigger, we can simply look to the right side of the graph. As x-values increase, the y-values increase. We can also write this symbolically, which we will see within the following sections.


    Limits can be expressed symbolically. Take this limit of g(x) = x2.

limit as x approaches infinity of g(x)

    The expression above is pronounced as Ďthe limit as x approaches infinity of g-of-x.í It means exactly the same meaning as was described within the last section, Limits as a Trend. We keep moving to the extreme right side of the graph and look to see what is happening to the y-values.

    By examining the graph, we can see that the y-values are getting greater and greater, without bound. This means that the y-values are moving toward infinity.

    Note: Infinity is not a number. It is a notion that means a value without bound or a value that will increase forever.


    Letís assume for this example that h(x) = x3. Let's calculate...

limit as x approaches infinity of h(x)

    This graph looks different than the graph of g(x). We saw that g(x) goes up on the right side of its graph and up on the left side of its graph. However, h(x) behaves differently. Here is the graph of h(x).

Quadratic Graph y=x^3

    It can be seen that h(x) goes up as we move to the extreme right-side of the graph. Conversely, as we move to the extreme left-side of the graph, h(x) goes down.

    Returning back to the notation of our problem, the limit states that we are looking at the graph as x approaches infinity. This means we move to the right side of the graph and observe the y-values, which are going up without bound. This means that the solution to our problem is infinity.

limit as x approaches infinity of h(x)




    Letís assume for this example that...

piecewise function k(x)

...and we wanted to calculate...

limit as x approaches negative infinity of k(x)

    For this example, we have to know something about Piecewise Functions. By graphing this piecewise function, we will better be able to evaluate the limit. Here is the graph of k(x).

Piecewise Function

    To calculate the limit of the function as the x-value approaches smaller and smaller values (negative infinity), we have to look at the far left side of the graph and observe what is happening to the y-values. The curve continuously moves down as we move to the left. This means the y-values are also getting smaller and smaller.

    This means...

limit as x approaches negative infinity of k(x) is negative infinity




    Take a look at the same function, k(x).

piecewise function k(x)

    We will calculate this limit.

limit as x approaches infinity of k(x)

    This time we will observe the graph (see previous section: Example 2) to determine what happens as the x-values get larger and larger. By looking at the extreme right side of the graph, we can see a trend.

    As we move to the right, the y-values neither increase nor decrease. They simply stay constant, at a y-value equal to 1.

    Therefore...

limit as x approaches infinity of k(x) is 1




    Try these lessons, which are related to the sections above.

    esson: Functions
    esson: Piecewise Functions
    esson: Limits of Ratios of Polynomials