Limits of Piecewise Functions
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Introduction

    This lesson page will inform you how to find various limits of piecewise functions. Here are the sections within this page:

    Before getting started, please familiarize yourself with our previous two lessons regarding limits. These lessons are essential for understanding the following sections within this lesson page.

    esson: Limits of the Extreme
    esson: Limits of Ratios of Polynomials

    Likewise, an essential part of understanding the limits of piecewise functions is to first understand piecewise functions. Review this lesson to learn more about piecewise functions.

    esson: Piecewise Functions


    When we determine a limit of a function, we attempt to see if there is a trend. Without actually evaluating the function at a specific x-value, we look to see what is happening to the y-values as we get closer to a certain x-value.

    If we were given the function f(x) that has been graphed below, we can determine the limit of the function as we approaches the x-value -1. If we are left of the x-value -1 and we move to the right, the y-values get larger. As we approach the x-value -1, the y-values get closer to 1. View the graph to confirm this fact.

graph of piecewise function

    Likewise, if we are right of the x-value -1 and we move to the left, we will notice how the y-values get smaller. As we approach the x-value -1, the y-values get closer to 1. No matter if we approach the x-value -1 from the left or the right, the y-value approaches 1.


    Within the last section (see Evaluating Limits), we performed two limits. The first limit can be written like so.

limit of f(x) as x approaches -1 from left

    It is pronounced as 'the limit of f(x) as x approaches -1 from the left. [The negative sign that is hovering on the -1 that looks like it is in the exponent position is the clue that we are being asked to approach the x-value -1 from the left side of -1.]     Similarly, we can write the limit as we approach the x-value -1 from the right side, like so.

limit of f(x) as x approaches -1 from right

    Again, the plus sign that is hovering in the exponent position mean that we are being asked to approach the x-value -1 but this time from the right side of -1.


    There are two cases that can happen when dealing with limits.

Case 1: The Limit Exists

    For some a-value, if...

definition of limit that exists

...then...

limit

Case 2: The Limit Does Not Exist

    For some a-value, if...

definition of limit that does not exist

...then...

limit does not exist

    DNE is shorthand for 'does not exist.'

    We already saw an example of case 1 (see the example within Limit Notation). So, we will examine a situation for case 2.

    The graph below is of g(x). We will use this graph of g(x) to determine the following limit.

limit of g(x) as x approaches -3

graph of piecewise function

    To determine this limit, we have to first consider both of these limits separately.

limit of g(x) as x approaches -3 from the left and the right

    First, we will consider this limit.

limit of g(x) as x approaches -3 from the left

    To calculate this limit, we have to approach the x-value -3 from the left side. We have to observe the y-values and see where they are headed as we move to the right, closer to the x-value -3. As we move closer and closer to the x-value -3, we can see that the y-values remain constant at -3.

    Now let's calculate the other limit.

limit of g(x) as x approaches -3 from the right

    We are being asked to approach the x-value -3, but from the right side of -3. As we move left along the curve, we eventually slide downward on the left side of the semi-circle. As we get closer to the x-value -3, we can see the y-value is moving toward 0.

    Notice how the two limits are not equal.

limit of g(x) as x approaches -3 from the left and the right

    Since the two limits are not equal, we say...

limit of g(x) as x approaches -3 does not exist

...which means the limit 'does not exist.'

    uiz: Limits of Piecewise Functions


    Try this interactive quiz, which are related to the sections above.

    uiz: Limits of Piecewise Functions


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

    esson: Functions
    esson: Piecewise Functions
    esson: Limits of the Extreme
    esson: Limits of Ratios of Polynomials
    esson: Limits: Numerical Approach