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

\[\lim_{x\to2.5^{-}}F(x)=5\]

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

\[\lim_{x\to2.5^{+}}F(x)=2,\]

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 closeto $a$ with $\boldsymbol{x<a}$, 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 \[\lim_{x\to a^{-}}f(x)=L.\]

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 $\boldsymbol{x>a}$, 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 \[\lim_{x\to a^{-}}f(x)=L.\]
  • 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,
    \[\lim_{x\to-2.5}f(x)\] means the limit of $f(x)$ as $x$ approaches $-2.5$ from both sides. In general \[\lim_{x\to-2.5}f(x)\neq\lim_{x\to2.5^{-}}f(x).\]

    Do not forget the + sign. For example,\[\lim_{x\to2.5}f(x)\]means the limit of $f(x)$ as $x$ approaches 2.5 from both sides, but\[\lim_{x\to2.5^{+}}f(x)\] 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 $y=\text{sgn}(x)$ 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 ${\displaystyle \lim_{x\to0^{-}}\text{sgn}(x)}$.

(b) Determine ${\displaystyle \lim_{x\to0^{+}}\text{sgn}(x)}$.

Solution

The graph of $y=\text{sgn}(x)$ is shown below. 

Graph of y = sgn(x).

 

(a) When $x$ is any negative number, the value of $\text{sgn}(x)$ is$-1$. Therefore
$$ \lim_{x\to0^{-}}\text{sgn}(x)=-1.$$

(b)  When $x$ is positive, the value of $\text{sgn}(x)$ is $1$. Therefore,
$$\lim_{x\to0^{+}}\text{sgn}(x)=1.$$

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

Theorem 1: ${\displaystyle \lim_{x\to a}f(x)}$ exists and is equal to $L$ if and only if ${\displaystyle \lim_{x\to a^{-}}f(x)}$ and ${\displaystyle \lim_{x\to a^{+}}f(x)}$ both exist and are equal to $L$. That is,

\[\lim_{x\to a}f(x)=L\quad\Longleftrightarrow\quad\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=L.\]

For instance, in the above example
$$ \lim_{x\to0^{-}}\text{sgn}(x)\neq \lim_{x\to0^{+}}\text{sgn}(x),$$
so ${\displaystyle \lim_{x\to0}\text{sgn}(x)}$ 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

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