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Workbook for Stewart’s "Calculus"

Cory Wilson

Section 1.5 The Limit of a Function

Subsection 1.5.1 Before Class

https://mymedia.ou.edu/media/1.5-1/1_42ce57lj
Figure 7. Pre-Class Video 1

Subsubsection 1.5.1.1 Motivating Example

Example 1.5.1.
Consider the function \(f(x) = x^3 -2x+1\text{.}\) A table of its values (to 6 decimal places) are given below:
\(x\) \(f(x)\) \(x\) \(f(x)\)
\(1.0\) \(0.000000\) \(3.0\) \(22.000000\)
\(1.5\) \(1.375000\) \(2.5\) \(11.625000\)
\(1.8\) \(3.232000\) \(2.2\) \(7.248000\)
\(1.9\) \(4.059000\) \(2.1\) \(6.061000\)
\(1.95\) \(4.514875\) \(2.05\) \(5.515125\)
\(1.99\) \(4.900599\) \(2.01\) \(5.100601\)
\(1.999\) \(4.990006\) \(2.001\) \(5.010006\)
Describe what is happening to the output values as \(x\) approaches 2.
Solution.
As \(x\to 2\text{,}\) \(f(x)\to 5\)

Subsubsection 1.5.1.2 The Limit

Definition 1.5.2. Intuitive Definition of a Limit.
Suppose \(f(x)\) is defined when \(x\) is near the input \(a\text{.}\) We write
\begin{equation*} \lim_{x\to a} f(x) = L \qquad f(x)\to L\text{ as }x\to a \end{equation*}
and say
“the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\text{,}\) equals \(L\)” or “\(f(x)\) approaches \(L\) as \(x\) approaches \(a\)”
if we can make the values of \(f(x)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(a\) (on either side of \(a\)) but not equal to \(a\) itself.
There is a more rigorous definition given in Section 1.7. One important aspect of the definition of a limit is that we do not require the function to exist at the input \(a\text{.}\)
A Word About Notation.
In mathematics, notation is the way you succinctly communicate your ideas. Notation is part of the “language” of math; proper notation ensures proper communication. Please pay very careful attention to how I am writing the notation in class.
Notation can be a confusing aspect of mathematics; that’s ok. It happens for Calculus I students and even for tenured professors. If you have any questions, please refer to your notes and/or ask me! I am happy to help you navigate things that are confusing or don’t make sense to you.
Example 1.5.3.
Draw three graphs so that the limit \(L\) is defined at the input \(a\text{,}\) but \(f(a)\) is distinct at all three points. This illustrates three possibilities when dealing with limits.
Solution.
The graph of \(y=x\) with a point located at \((a,f(a))\)
The graph of \(y=x\) with the point located at \((a,f(a))\) removed
The graph of \(y=x\) with a point located at \((a,f(a))\) removed and moved above
Example 1.5.4.
Consider \(f(x) = \dfrac{x-1}{x^2-1}\text{.}\) What is \(\ds \lim_{x\to 1} f(x)\text{?}\)
  1. Fill out the table below (using a calculator) to six decimal places. What do you think the limit will be? Why?
    \(x\) \(f(x)\) \(x\) \(f(x)\)
    0 2
    0.5 1.5
    0.75 1.25
    0.99 1.01
  2. Simplify the function. Can you determine the limit from your simplified version?
Solution.
  1. \(x\) \(f(x)\) \(x\) \(f(x)\)
    0 1 2 0.333333
    0.5 0.666667 1.5 .4
    0.75 0.571429 1.25 0.444444
    0.99 0.502513 1.01 0.497512
  2. The function simplifies to \(\dfrac{1}{x+1}\text{,}\) so at \(x=1\) we would expect an answer of \(\dfrac{1}{2}\)

Subsection 1.5.2 Pre-Class Activities

Example 1.5.5.

What if instead we tweaked the function in the previous example to be
\begin{equation*} g(x) = \begin{cases}\dfrac{x-1}{x^2-1} \amp x\neq 1 \\ 2 \amp x = 1 \end{cases} \end{equation*}
What will happen to the limit? Explain your answer.
Solution.
Nothing changes; the limit cares about "expected" output, not actual output.

Example 1.5.6.

  1. Translate the expression \(\ds \lim_{x\to 3} f(x) = 7\) from math into English. Explain the sentence you wrote.
  2. Is it possible for \(f(3)\) to be \(-10\) instead of 7? Why or why not?
Solution.
  1. As \(x\) approaches 3, the output approaches 7. This means that when the inputs get close to 3, the outputs get closer to 7
  2. Yes. Limits are concerned with expected output, not actual output; a function can have a different output than its limit at the same point.

Subsection 1.5.3 In Class

Subsubsection 1.5.3.1 One-Sided Limits

When we developed the intuition of a limit, we looked at input values which were smaller and larger than our desired input. These values are to the left and right of our input; secretly, we were taking one-sided limits.
Definition 1.5.7. One-Sided Limits.
Suppose \(f(x)\) is defined when \(x\) is near the input \(a\text{.}\) We write
\begin{equation*} \lim_{x\to a^+} f(x) = L \end{equation*}
and say
“the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\) from the right, equals \(L\)” or “\(f(x)\) approaches \(L\) as \(x\) approaches \(a\) from the right”
if we can make the values of \(f(x)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(a\) (using \(x \lt a\)). We write
\begin{equation*} \lim_{x\to a^-} f(x) = L \end{equation*}
and say
“the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\) from the left, equals \(L\)” or “\(f(x)\) approaches \(L\) as \(x\) approaches \(a\) from the left”
if we can make the values of \(f(x)\) arbitrarily close to \(L\) by restricting \(x\) to be sufficiently close to \(a\) (using \(x \gt a\))
Existence of a Limit.
A function \(f(x)\) has a limit at \(x = c\) if and only if its one-sided limits exist and agree.
Example 1.5.8.
Use the graph of \(g\) to answer the following:
This graph is not filled in at \((3,7)\text{,}\) is filled in at \((3,5)\text{,}\) and continues.
  1. \(\displaystyle \ds \lim_{x\to 4^-} g(x)=\)
  2. \(\displaystyle \ds \lim_{x\to 4^+} g(x)=\)
  3. \(\displaystyle \ds \lim_{x\to 4} g(x)=\)
  4. \(\displaystyle \ds \lim_{x\to 3^-} g(x)=\)
  5. \(\displaystyle \ds \lim_{x\to 3^+} g(x)=\)
  6. \(\displaystyle \ds \lim_{x\to 3} g(x)=\)
Solution.
  1. \(\displaystyle \ds \lim_{x\to 4^-} g(x)=2\)
  2. \(\displaystyle \ds \lim_{x\to 4^+} g(x)=2\)
  3. \(\displaystyle \ds \lim_{x\to 4} g(x)=2\)
  4. \(\displaystyle \ds \lim_{x\to 3^-} g(x)=6\)
  5. \(\displaystyle \ds \lim_{x\to 3^+} g(x)=5\)
  6. \(\displaystyle \ds \lim_{x\to 3} g(x) \text{ does not exist}\)
Example 1.5.9.
Use a calculator to examine the limit behavior of the function \(r(p) = \dfrac{p^2-64}{p+8}\) at \(p = -8\text{.}\) Record your answers will full decimal accuracy, and round your final answer to the thousandths place, if necessary.
\(p\) \(r(p)\)
\(-8.1\) \(\)
\(-8.01\) \(\)
\(-8.001\) \(\)
\(-8.0001\) \(\)
\(-8.00001\) \(\)
\(\ds \lim_{p\to -8^-}r(p) =\) \(\)
\(p\) \(r(p)\)
\(-7.9\) \(\)
\(-7.99\) \(\)
\(-7.999\) \(\)
\(-7.9999\) \(\)
\(-7.99999\) \(\)
\(\ds \lim_{p\to -8^+}r(p) =\) \(\)
\begin{equation*} \ds \lim_{p\to -8}r(p)= \end{equation*}
Solution.
\(p\) \(r(p)\)
\(-8.1\) \(-16.1\)
\(-8.01\) \(-16.01\)
\(-8.001\) \(-16.001\)
\(-8.0001\) \(-16.0001\)
\(-8.00001\) \(-16.00001\)
\(\ds \lim_{p\to -8^-}r(p) = \) \(-16 \)
\(p\) \(r(p)\)
\(-7.9\) \(-15.9\)
\(-7.99\) \(-15.99\)
\(-7.999\) \(-15.999\)
\(-7.9999\) \(-15.9999\)
\(-7.99999\) \(-15.99999\)
\(\ds \lim_{p\to -8^+}r(p) = \) \(-16\)
\begin{equation*} \ds \lim_{p\to -8}r(p)= -16 \end{equation*}

Subsubsection 1.5.3.2 Infinite Limits

Example 1.5.10.
Estimate \(\ds \lim_{x\to 0} -\dfrac{1}{x^2}\text{.}\)
\(x\) \(-\dfrac{1}{x^2}\)
\(-0.1\) \(\)
\(-0.01\) \(\)
\(-0.001\) \(\)
\(-0.0001\) \(\)
\(-0.00001\) \(\)
\(\ds \lim_{x\to 0^-}\lrpar{-\dfrac{1}{x^2}} =\) \(\)
\(x\) \(-\dfrac{1}{x^2}\)
\(0.1\) \(\)
\(0.01\) \(\)
\(0.001\) \(\)
\(0.0001\) \(\)
\(0.00001\) \(\)
\(\ds \lim_{x\to 0^+}\lrpar{-\dfrac{1}{x^2}} =\) \(\)
\begin{equation*} \lim_{x\to 0}\lrpar{-\dfrac{1}{x^2}}= \end{equation*}
Solution.
\(x\) \(-\dfrac{1}{x^2}\)
\(-0.1\) \(-100\)
\(-0.01\) \(-10000\)
\(-0.001\) \(-1000000\)
\(-0.0001\) \(-100000000\)
\(-0.00001\) \(-10000000000\)
\(\ds \lim_{x\to 0^-}\lrpar{-\dfrac{1}{x^2}} =\) \(-\infty\)
\(x\) \(-\dfrac{1}{x^2}\)
\(0.1\) \(-100\)
\(0.01\) \(-10000\)
\(0.001\) \(-1000000\)
\(0.0001\) \(-100000000\)
\(0.00001\) \(-10000000000\)
\(\ds \lim_{x\to 0^+}\lrpar{-\dfrac{1}{x^2}} =\) \(-\infty\)
\begin{equation*} \lim_{x\to 0}\lrpar{-\dfrac{1}{x^2}}= -\infty \end{equation*}
Question 1.5.11.
What is different about these outputs than in our previous examples?
Solution.
Answers vary
Notation.
We use the notation \(\infty\) or \(-\infty\) to indicate a value which gets arbitrarily large. IT IS NOT A NUMBER!
Definition 1.5.12. Intuitive Definition of an Infinite Limit.
Let \(f\) be a function defined on both sides of \(a\text{,}\) except possibly at \(a\) itself. If the values of \(f(x)\) can be made arbitrarily large, by taking \(x\) sufficiently close to \(a\) (but not equal to \(a\)). We denote this as
\begin{equation*} \lim_{x\to a} f(x) = \infty \end{equation*}
If the values can be made arbitrarily negative, we denote this as
\begin{equation*} \lim_{x\to a} f(x) = -\infty \end{equation*}
One-sided limits are defined similarly.
Example 1.5.13.
Sketch an example of each of the following:
  1. \(\displaystyle \displaystyle \lim_{x\to a^-} f(x) = \infty\)
  2. \(\displaystyle \displaystyle \lim_{x\to a^+} f(x) = \infty\)
  3. \(\displaystyle \displaystyle \lim_{x\to a} f(x) = \infty\)
  4. \(\displaystyle \displaystyle \lim_{x\to a^-} f(x) = -\infty\)
  5. \(\displaystyle \displaystyle \lim_{x\to a^+} f(x) = -\infty\)
  6. \(\displaystyle \displaystyle \lim_{x\to a} f(x) = -\infty\)
Solution.
  1. A graph with vertical asymptote at \(x=a\text{,}\) increasing as \(x\to a^-\)
  2. A graph with vertical asymptote at \(x=a\text{,}\) increasing as \(x\to a^+\)
  3. A graph with vertical asymptote at \(x=a\text{,}\) increasing as \(x\to a^-\) and as \(x\to a^+\)
  4. A graph with vertical asymptote at \(x=a\text{,}\) decreasing as \(x\to a^-\)
  5. A graph with vertical asymptote at \(x=a\text{,}\) decreasing as \(x\to a^+\)
  6. A graph with vertical asymptote at \(x=a\text{,}\) decreasing as \(x\to a^-\) and \(x\to a^+\)
Definition 1.5.14. Vertical Asymptote.
The vertical line \(x = a\) is called a vertical asymptote of the curve \(y = f(x)\) if at least one of the following statements is true:
\begin{equation*} \lim_{x\to a} f(x) = \infty \qquad \lim_{x\to a^-} f(x) = \infty \qquad \lim_{x\to a^+}f(x) = \infty \end{equation*}
\begin{equation*} \lim_{x\to a} f(x) = -\infty \qquad \lim_{x\to a^-} f(x) = -\infty \qquad \lim_{x\to a^+} f(x) = -\infty \end{equation*}
Example 1.5.15.
Find the left- and right-hand limits for the function \(f(x) = \dfrac{4x}{x-6}\) at \(x = 6\text{.}\)
Solution.
\(\ds \lim_{x\to 6^-} \dfrac{4x}{x-6} = -\infty\) and \(\ds \lim_{x\to 6^+} \dfrac{4x}{x-6} = \infty\)
Example 1.5.16.
In the theory of relativity, the mass of a particle with velocity \(v\) is given by \(m = \dfrac{m_0}{\sqrt{1-v^2/c^2}}\text{,}\) where \(m_0\) is the mass of the particle at rest and \(c\) is the speed of light. What happens as \(v\to c^-\text{?}\)
Solution.
As \(v\to c^-\text{,}\) \(m\to \infty\text{,}\) since we create a vertical asymptote in the function.

Subsection 1.5.4 After Class Activities

Example 1.5.17.

For the function \(g\) whose graph is given below, state the value of each quantity (if it exist). If it does not exist, explain why.
  1. \(\displaystyle \ds \lim_{t\to 0^-} g(t)\)
  2. \(\displaystyle \ds \lim_{t\to 0^+} g(t)\)
  3. \(\displaystyle \ds \lim_{t\to 0} g(t)\)
  4. \(\displaystyle \ds \lim_{t\to 2^-} g(t)\)
  5. \(\displaystyle \ds \lim_{t\to 2^+} g(t)\)
  6. \(\displaystyle \ds \lim_{t\to 2} g(t)\)
  7. \(\displaystyle g(2)\)
  8. \(\displaystyle \ds \lim_{t\to 4} g(t)\)
Solution.
  1. \(\displaystyle -1\)
  2. \(\displaystyle -2\)
  3. \(\displaystyle DNE\)
  4. \(\displaystyle 2\)
  5. \(\displaystyle 0\)
  6. \(\displaystyle DNE\)
  7. \(\displaystyle 1\)
  8. \(\displaystyle 3\)

Example 1.5.18.

Sketch a graph of an example of a function \(f\) that satisfies the given conditions: \(\ds \lim_{x\to 0} f(x) = 1\text{,}\) \(\ds \lim_{x\to 3^-} f(x) = -2\text{,}\) \(\ds \lim_{x\to 3^+} f(x) = 2\text{,}\) \(f(0) = -1\text{,}\) \(f(3) = 1\text{.}\)
Solution.
Answers vary

Example 1.5.19.

Determine the limits for the following functions:
  1. \(\displaystyle \ds \lim_{x\to 5^+} \dfrac{x+1}{x-5}\)
  2. \(\displaystyle \ds \lim_{x\to 5^-} \dfrac{x+1}{x-5}\)
  3. \(\displaystyle \ds \lim_{x\to 2\pi^+} x\csc x\)
Solution.
  1. \(\displaystyle \infty\)
  2. \(\displaystyle -\infty\)
  3. \(\displaystyle \infty\)
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