4.3 One-Sided Limits

Consider the function F(x) whose graph is shown in Figure 1. If we take x values closer and closer to 2.5, but less than 2.5, F(x) gets closer and closer to 5. In other words, when x approaches 2.5 through the values less than 2.5, F(x) approaches 5. We express this by saying that “the limit of F(x) as x approaches 2.5 from the left is 5” or “the left-hand limit of F(x) as x approaches 2.5 is 5.” The notation for this is

limx→2.5−F(x)=5

The minus sign that is written after 2.5 means x approaches 2.5 from the left.

Figure 1: Graph of y=F(x).

 

Now consider the case in which x takes on the values close to 2.5 but larger than 2.5. As x approaches 2.5 from the right, F(x) approaches 2. Symbolically we write

limx→2.5+F(x)=2,

and say “the limit of F(x) as x approaches 2.5 from the right is 2” or “the right-hand limit of F(x) as x approaches 2.5 is 2.”

In this example, F(x) is defined at x=2.5, but the value of F(2.5) has no bearing on the left-hand or right-hand limit of F(x). Even if we remove x=2.5 from the domain of F(x) (that is, if F(x) were not defined at x=2.5), the left-hand and right-hand limits would remain the same.

Definition 1 (left-hand limit): If we can make the values of f(x) as close as we please to a number L by taking x sufficiently close to a with x<a, we say “the limit of f(x) as x approaches a from the left is L” or “the left-hand limit of f(x) as x approaches a is L” and write limx→a−f(x)=L.

Similarly

Definition 2 (right-hand limit): If we can make the values of f(x) as close as we please to a number L by taking x sufficiently close to a with x>a, we say “the limit of f(x) as x approaches a from the left is L” or “the left-hand limit of f(x) as x approaches a is L” and write limx→a−f(x)=L.
  • For one-sided limits always put the plus or minus sign after a.
    Do not forget them and do not put them before a. For example,
    limx→−2.5f(x) means the limit of f(x) as x approaches −2.5 from both sides. In general limx→−2.5f(x)≠limx→2.5−f(x).

    Do not forget the + sign. For example,limx→2.5f(x)means the limit of f(x) as x approaches 2.5 from both sides, butlimx→2.5+f(x) means the limit of f(x) as x approaches 2.5 from the right.

Once again consider the sign function:

Example 1

Consider the function y=sgn(x) defined by

\large \text{sgn}(x)=\begin{cases} 1 & \text{if } x>0\\ 0 & \text{if }x=0\\ -1 & \text{if } x<0 \end{cases}

(a) Determine limx→0−sgn(x).

(b) Determine limx→0+sgn(x).

Solution

The graph of y=sgn(x) is shown below. 

Graph of y = sgn(x).

 

(a) When x is any negative number, the value of sgn(x) is −1. Therefore
limx→0−sgn(x)=−1.

(b)  When x is positive, the value of sgn(x) is 1. Therefore,
limx→0+sgn(x)=1.

By comparing the definitions of one-sided limits and regular (or two-sided) limits, we see the following is true.

Theorem 1: limx→af(x) exists and is equal to L if and only if limx→a−f(x) and limx→a+f(x) both exist and are equal to L. That is,

limx→af(x)=L⟺limx→a−f(x)=limx→a+f(x)=L.

For instance, in the above example
limx→0−sgn(x)≠limx→0+sgn(x),
so limx→0sgn(x) does not exist as we learned in the Section on the Concept of a Limit.

Example 2

The graph of f(x) is shown in the following figure. Evaluate the left-hand, right-hand, and limit (also two-sided or regular limit) of f(x) as x approaches −4,−3,−1,1,2, and 4.

Solution