We introduced the concept of a limit gently, approximating their values graphically and numerically. Next came the rigorous definition of the limit, along with an admittedly tedious method for evaluating them. Section 1.3 gave us tools (which we call theorems) that allow us to compute limits with greater ease. Chief among the results were the facts that polynomials and rational, trigonometric, exponential and logarithmic functions (and their sums, products, etc.) all behave “nicely.” In this section we rigorously define what we mean by “nicely.”
In this section we explore in depth the concepts behind Item 1 by introducing the one-sided limit. We begin with formal definitions that are very similar to the definition of the limit given in Section 1.2, but the notation is slightly different and “” is replaced with either “” or “.”
Definition1.4.1.One Sided Limits: Left- and Right-Hand Limits.
Left-Hand Limit
Let be a function defined on for some and let be a real number. The statement that the limit of , as approaches from the left, is , (alternatively, that the left-hand limit of at is ) is denoted by
Let be a function defined on for some and let be a real number. The statement that the limit of , as approaches from the right, is , (alternatively, that the right-hand limit of at is ) is denoted by
Practically speaking, when evaluating a left-hand limit, we consider only values of “to the left of ,” i.e., where . The admittedly imperfect notation is used to imply that we look at values of to the left of . The notation has nothing to do with positive or negative values of either or . It’s more like you are adding very small negative values to to get values for . A similar statement holds for evaluating right-hand limits; there we consider only values of to the right of , i.e., . We can use the theorems from previous sections to help us evaluate these limits; we just restrict our view to one side of .
Graph of the piecewise function . There are two line segments: for we have a line segment with positive slope, and for we have a line segment with negative slope.
The line segment with a positive slope starts at the point and ends at . The line segment with a negative slope starts at and ends at .
The start and end points of the line segment with a positive slope are solid dots, indicating that those points are part of the graph. The start and end points of the line segment with a negative slope are hollow dots. This tells that although the second line segment gets arbitrarily close to the points and , these points are not part of the graph.
Since is close to when is close to 1, but , while is close to when is close to 1, but , we can conclude that the left and right hand limits are different.
For these problems, the visual aid of the graph is likely more effective in evaluating the limits than using itself. Therefore we will refer often to the graph.
As goes to from the left, we see that is approaching the value of .
Therefore .
As goes to from the right, we see that is approaching the value of . Recall that it does not matter that there is an “open circle” there; we are evaluating a limit, not the value of the function.
Therefore .
The limit of as approaches does not exist, as discussed in Section 1.1. The function does not approach one particular value, but two different values from the left and the right.
Using the definition, and by looking at the graph, we see that .
As goes to from the right, we see that is approaching . Therefore . Note we cannot consider a left-hand limit at as is not defined for values of .
Using the definition and the graph, .
As goes to from the left, we see that is approaching the value of .
Therefore .
The graph and the definition of the function show that is not defined.
Note how the left- and right-hand limits were different at . This, of course, causes the limit to not exist. The following theorem states what is fairly intuitive: the limit exists precisely when the left- and right-hand limits are equal.
The phrase “if, and only if” means the two statements are equivalent: they are either both true or both false. If the limit equals , then the left and right hand limits both equal . If the limit is not equal to , then at least one of the left and right-hand limits is not equal to (it may not even exist).
One thing to consider in Examples 1.4.3–1.4.10 is that the value of the function may/may not be equal to the value(s) of its left/right-hand limits, even when these limits agree.
In this example, we evaluate each expression using just the definition of , without using a graph as we did in the previous example.
As approaches from the left, we consider a limit where all -values are less than . This means we use the “” piece of the piecewise-defined function . As the -values near , approaches ; that is, approaches .
Therefore
A concise mathematical presentation of the above argument could be written as follows:
for properties of limits
As approaches from the right, we consider a limit where all -values are greater than . This means we use the “” piece of . As the -values near , approaches ; that is, we see that again approaches .
Therefore .
Once again, we can present our work computationally as follows:
for properties of limits
The limit of as approaches exists and is , as approaches from both the right and left.
Therefore .
Neither piece of is defined for the -value of ; in other words, is not in the domain of . Therefore is not defined.
As approaches from the right, we consider a limit where all -values are greater than . This means we use the piece of . As the -values near , approaches ; that is, approaches .
So .
is not defined as is not in the domain of .
As approaches from the left, we consider a limit where all -values are less than . This means we use the piece of . As the -values near , nears ; that is, approaches .
So .
is not defined as is not in the domain of .
We can confirm our analytic result by consulting the graph of shown in Figure 1.4.7. Note the open circles on the graph at , and , where is not defined.
Graph of from Example 1.4.6. The graph consists of two parts. The first part is a line that starts at the point and ends at . The second part is a curve that starts at and ends at . The points ,, and are all marked with hollow dots, indicating that although the graph gets close to these points, they are not part of the graph.
The function is undefined for , but the graph shows that approaches the same value (namely, 1) from both the left and the right, allowing us to conclude that exists, and is equal to 1.
The graph of in Example 1.4.8. The graph is a parabola, opening upward, plotted from to , with its vertex at . However, there is a hole in the graph at the vertex, indicated by a hollow dot. The function is still defined at , because there is a solid dot at . This shows that , but the limit of as approaches is .
It is clear by looking at the graph that both the left- and right-hand limits of , as approaches , are . Thus it is also clear that the limit is ; i.e., . It is also clearly stated that .
Graph of a piecewise-defined function, on the interval . For values between and (inclusive) the graph is an upward curved parabola. For values to (inclusive) the graph is a straight line with a negative slope.
The parabola and the line meet at the point .
There are three solid dots plotted on the graph at the points, ,, and , to indicate where each part of the graph begins and ends.
Only in Example 1.4.10 do both the function and the limit exist and agree. This seems “nice;” in fact, it seems “normal.” This is in fact an important situation which we explore in Section 1.5 entitled “Continuity.” In short, a continuous function is one in which when a function approaches a value as (i.e., when ), it actually attains that value at . Such functions behave nicely as they are very predictable.