3.7: Left and right limits and derivatives; limits involving infinity.

Suppose \(F\) is a function defined for all \(x\). We give meaning to the following symbols.

\[\lim _{x \rightarrow a^{-}} F(x) \quad \text { and } \quad F^{-}(a) \text {. }\]

Having done so, we ask you to define

\[\lim _{x \rightarrow a^{+}} F(x) \quad \text { and } \quad F^{+}(a) \text {. }\]

Next we will define limits involving infinity,

\[\lim _{x \rightarrow \infty} F(x) \quad \text { and } \quad \lim _{x \rightarrow a^{-}} F(x) = \infty \text {, }\]

and ask you to define

\[\lim _{x \rightarrow - \infty} F(x) \quad \text { and } \quad \lim _{x \rightarrow a^{+}} F(x) = \infty \text {, }\]

In reading the following two definitions, it will be helpful to study the graphs in Figure 3.7.1 and assume \(a = 3\).

Figure \(\PageIndex{1}\): Graphs of functions \(F, G\), and \(H\) that illustrate Definitions 3.7.1 and 3.7.2.

Definition 3.7.1 Left hand limit and derivative

Suppose \(F\) is a function defined for all numbers \(x\) except perhaps for a number \(a\) and \(L\) is a number.

\[\text { The statement that } \lim _{x \rightarrow a^{-}} F(x)=L \quad \text { means that }\]

if \(\epsilon\) is a positive number there is a positive number \(\delta\) such that if \(x\) is in the domain of \(F\) and \(a − \delta < x < a\), then \(|F(x) − L| < \epsilon\).

If \(F\) is defined at \(a\), then \(F ^{\prime -} (a)\) is defined by

\[F^{\prime-}(a)=\lim _{x \rightarrow a^{-}} \frac{F(x)-F(a)}{x-a} \label{3.37}\]

For convenience, in the definitions we have assumed that \(F\) is defined for all numbers except perhaps \(a\); in Definition 3.7.1, for example, it would be sufficient to assume that for every number \(b\) less than \(a\), the domain of \(F\) contains a number between \(b\) and \(a\). The condition \(a − \delta < x < a\) restricts \(x\) to being 'to the left' of \(a ~ (x < a)\) and within \(\delta\) of \(a\). In Figure 3.7.1A, \(\lim _{x \rightarrow 3^{-}} F(x) = 1\). In Figure 3.7.1C, \(H ^{\prime -} (3) = -1/3\). This may help resolve some dispute as to whether \(H\) has a tangent at (3,2) that was mentioned early in this chapter, Explore 3.1.2.

Definition 3.7.2 Limits involving infinity

Suppose \(F\) is a function defined for all numbers \(x\) except perhaps for a number \(a\) and \(L\) is a number.

\[\text{The statement that} \quad \lim _{x \rightarrow \infty} F(x)=L \quad \text{means that}\]

if \(\epsilon\) is a positive number there is a number \(M\) so that if \(x > M, |F(x)-L| < \epsilon\).

The half-line \(y = L\) for \(x > 0\) is said to be a horizontal asymptote of \(F\).

\[\text{The statement that} \quad \lim _{x \rightarrow a} F(x)= \infty \quad \text{means that}\]

if \(M\) is a number there is a positive number \(\delta\) so that if \(0 < |a − x| < \delta, F(x) > M\).

In Figure 3.7.1B, \(\lim _{x \rightarrow \infty} G(x)= 2\), and \(\lim _{x \rightarrow 3} G(x)\) does not exist. However, \(\lim _{x \rightarrow 3^{-}} G(x) = \infty\)

Explore 3.7.1 Refer to Figure 3.7.1. For each part below, either evaluate the expression or explain why it is not defined.

1. \(\lim _{x \rightarrow 3^{+}} F(x)\)
2. \(\lim _{x \rightarrow 3} F(x)\)
3. \(\lim _{x \rightarrow 3^{+}} G(x)\)
4. \(F ^{\prime -} (3)\)
5. \(H ^{\prime +} (3)\)
6. \(H ^{\prime} (3)\)
7. \(\lim _{x \rightarrow \infty} F(x)\)
8. \(\lim _{x \rightarrow - \infty} F(x)\)
9. \(\lim _{x \rightarrow - \infty} H(x)\)

Explore 3.7.2 Write definitions for:

1. \(\lim _{x \rightarrow a^{+}} F(x) = L\)
2. \(F ^{\prime +} (a)\)
3. \(\lim _{x \rightarrow - \infty} F(x) = L\)
4. \(\lim _{x \rightarrow a^{+}} F(x) = - \infty\)

Explore 3.7.3 Attention: Solving this problem may require a significant amount of thought.

We say that \(\lim _{x \rightarrow a^{-}} F(x)\) exists if either

\[\lim _{x \rightarrow a^{-}} F(x)=-\infty, \quad \lim _{x \rightarrow a^{-}} F(x)=\infty, \quad \text { or for some number } L \quad \lim _{x \rightarrow a^{-}} F(x)=L\]

Is there a function, \(F\), defined for all numbers \(x\) such that

\[\lim _{x \rightarrow 1^{-}} F(x) \quad \text{does } \textbf{not} \text{ exist}.\]

Proof of the following theorem is only technical and is omitted.

Theorem 3.7.1 Suppose \((p, q)\) an open interval containing a number \(a\) and \(F\) is a function defined on \((p, q)\) excepts perhaps at \(a\) and \(L\) is a number.

\[\lim _{x \rightarrow a} F(x)=L \quad \text { if and only if both } \quad \lim _{x \rightarrow a^{-}} F(x)=L \quad \text { and } \quad \lim _{x \rightarrow a^{+}} F(x)=L \text {. } \label{3.38}\]

Furthermore, if \(F\) is defined at \(a\)

\[F^{\prime}(a) \quad \text { exists if and only if } \quad F^{\prime-}(a)=F^{\prime+}(a), \quad \text { in which case } \quad F^{\prime}(a)=F^{\prime-}(a) \text {. } \label{3.39}\]

 

Exercises for Section 3.7 Left and right limits and derivatives; limits involving infinity.

Exercise 3.7.1 For each of the graphs in Figure 3.7.1 of a function, \(F\), answer the questions or assert that there is no answer available.

1. What does \(F(t)\) approach as t approaches 3?
2. What does \(F(t)\) approach as t approaches \(3^{-}\)?
3. What does \(F(t)\) approach as t approaches \(3^{+}\)?
4. What is \(F(3)\)?
5. Use the limit symbol to express answers to a. - c.

Figure for Exercise 3.7.1 Graphs of three functions for Exercise 3.7.1.

Exercise 3.7.2 Let functions \(D, E, F, G\), and \(H\) be defined by

\[\begin{aligned}
&D(x)=|x| \quad \text { for all } x \\
&E(x)=\left\{\begin{array}{rll}
-1 & \text { for } & x<0 \\
0 & \text { for } & x=0 \\
1 & \text { for } & 0<x
\end{array}\right. \\
&F(x)=\left\{\begin{array}{r}
x^{2} \text { for } x<0 \\
\sqrt{x} \text { for } x \geq 0
\end{array}\right. \\
&G(x)=\left\{\begin{array}{rll}
1 & \text { for } & x \neq 0 \\
2 & \text { for } & x=0
\end{array}\right. \\
&H(x)=\left\{\begin{array}{r}
x \text { for } x<0 \\
x^{2} \text { for } x \geq 0
\end{array}\right.
\end{aligned}\]

1. Sketch the graphs of \(D, E, F, G\), and \(H\).
2. Let \(K\) be either of \(D, E, F, G\), or \(H\). Evaluate the limits or show that they do not exist.
   1. \(\lim _{x \rightarrow 0^{-}} K(x)\)
   2. \(\lim _{x \rightarrow 0^{+}} K(x)\)
   3. \(\lim _{x \rightarrow 0} K(x)\)
   4. \(K ^{\prime -} (0)\)
   5. \(K ^{\prime +} (0)\)
   6. \(K ^{\prime} (0)\)

Exercise 3.7.3 Define the term 'tangent from the left'. (We assume you would have a similar definition for 'tangent from the right'; no need to write it.) What is the relation between tangent from the left, tangent from the right, and tangent?