Short and Sweet Calculus

2.2 One-sided limits

Consider the function \(F(x)\) whose graph is shown in Figure 2.2. 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.

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.

2.2. (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 \(\boldsymbol{x<a}\), 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

2.3. (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

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

Example 2.1. Consider the function \(y=\text{sgn}(x)\) defined by \[{\rm sgn}(x)=\left\{ \begin{tabular}{ll} \ensuremath{1} & if \ensuremath{x>0}\\ 0 & if \ensuremath{x=0}\\ \ensuremath{-1} & if \ensuremath{x<0} \end{tabular}\right.\] (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. (a) When \(x\) is any negative number, the value of \(\text{sgn}(x)\) is \(-1\). Therefore \({\displaystyle \lim_{x\to0^{-}}\text{sgn}(x)}=-1\).
(b) When \(x\) is positive, the value of \(\text{sgn}(x)\) is \(1\). Therefore, \({\displaystyle \lim_{x\to0^{+}}\text{sgn}(x)=1}\).

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

2.1. \({\displaystyle \lim_{x\to a}f(x)}\) 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\({\displaystyle \lim_{x\to0^{-}}\text{sgn}(x)}\neq{\displaystyle \lim_{x\to0^{+}}\text{sgn}(x)}\), so \({\displaystyle \lim_{x\to0}\text{sgn}(x)}\) does not exist.


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