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Handout Daily Prep 3.2 - Using Derivatives to Evaluate Limits

Section Overview

As we move on to Chapter 3, the theme of our work will focus on how we can apply the meaning of the derivative to solve important problems. Interestingly, although we use limits to define the derivative itself, it turns out that the derivative can be a useful tool in evaluating challenging limits of a certain type.

The topics discussed in this section are: Indeterminate forms of type “0/0”. Local linearization and L’Hôpital’s Rule. Infinite limits and limits at infinity. Asymptotes. Indeterminate forms of type “$\infty/\infty$”. Indeterminate products of the form “$0 \cdot \infty$”. Indeterminate powers of the form “$\infty^{0}$”, “$1^{\infty}$”, “$0^{0}$”.

Section Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.

State what it means to say that a limit has an indeterminate form and how such forms arise in the limit definition of the derivative.
State L’Hôpital’s Rule.
Explain how L’Hôpital’s Rule is used to calculate limits having an indeterminate form.
Explain what the expression $\displaystyle \lim_{x \rightarrow \infty}f(x) = L$ means in plain English.
State the “$\infty$” version of L’Hôpital’s Rule.

Section To prepare for class

Complete all actions listed below. Respond to the questions highlighted with Submit.

Read motivating questions and the introduction to section 3.2 (up until Preview Activity 3.2.1).
Ask Copilot “What is meant by an indeterminant form of a limit?”
📝 [Submit] Do Preview Activity 3.2.1.
- (Optional) Watch video solution to Preview Activity 3.2.1.
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Read section 3.2.2.
📝 [Submit] Do the following construction (see Figure 75) in GeoGebra. Note how the red graph of the quotient $h(x)$ suggests that $\displaystyle \lim_{x\rightarrow 1}\frac{x^{5}+x-2}{x^{2}-1}= 3\text{.}$ The value of $b$ is what is computed via L’Hôpital’s rule.

Figure 75. A GeoGebra construction used to investigate a limit that is indeterminant.

Repeat this construction to analyze the limit found in Activity 3.2.2(a). That is, use $f(x) = \ln(1+x)\text{,}$ $g(x) = x\text{,}$ and investigate the limit of $f(x)/g(x)$ as $x \rightarrow 0\text{.}$ Submit screenshots as needed.
📝 [Submit] Do form a table of values using a spreadsheet (Excel or Google Sheets) to form a hypothesis regarding the value of $\displaystyle \lim_{x \rightarrow 0}\frac{e^{2x}-1}{x}\text{.}$ Values of $x$ that are both negative and positive near zero should be used. Then, repeat the GeoGebra exploration above to see if the same value for the limit emerges. [Yes, you should get 2.] Sample output is shown in Figure 76. Submit screenshots as needed.

Figure 76. A spreadsheet can be used to form a hypothesis about the value of a limit.
Do the following problem.
1. We attempt to evaluate the value of $\displaystyle \lim_{x \rightarrow 0}\frac{e^{2x}-1}{x}\text{.}$ Define $f(x) = e^{2x}-1$ (the numerator) and define $g(x) = x$ (the denominator). Then $\displaystyle \lim_{x \rightarrow 0}\frac{e^{2x}-1}{x}= \lim_{x \rightarrow 0}\frac{f(x)}{g(x)}.$
  1. Does $\displaystyle \lim_{x \rightarrow 0}\frac{f(x)}{g(x)}= \frac{f(0)}{g(0)}\text{?}$ Why or why not?
  2. Find the linear approximation $L(x)$ to $f(x)$ at $a=0\text{.}$
  3. Find the linear approximation $M(x)$ to $g(x)$ at $a=0\text{.}$
  4. Compute $\displaystyle \lim_{x \rightarrow 0}\frac{L(x)}{M(x)}\text{.}$ How is this related to $\displaystyle \frac{f'(0)}{g'(0)}\text{?}$
  5. For $x$ near 0,
    
    \begin{equation*} \frac{f(x)}{g(x)}= \frac{e^{2x}-1}{x}\approx \frac{2x}{x}= \frac{L(x)}{M(x)}= \frac{f'(0)}{g'(0)}. \end{equation*}
    
    As $x \rightarrow 0\text{,}$ the approximation becomes better. This tells us $\displaystyle \lim_{x \rightarrow 0}\frac{e^{2x}-1}{x}={\underline{\hspace{30mm}}}\text{.}$
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Watch video Quick Review - L’Hopital’s Rule (3:27).
Watch video Using L’Hopital’s Rule with Zero (7:26).
Do Exercises 1-2 in section 3.2.4.

Checkpoint 77. Evaluate the Limit.

Match each limit with its value.

\begin{equation*} \lim_{x\to 0}\frac{\sin(x)}{x} \end{equation*}
\begin{equation*} \lim_{x\to 0}\frac{\ln(1+x)}{x} \end{equation*}
$1$
\begin{equation*} \lim_{x\to 0}\frac{e^{2x}-1}{x} \end{equation*}
$2$
\begin{equation*} \lim_{t\to 0}\frac{e^t-1-t}{t^2} \end{equation*}
$\frac{1}{2}$

Section After class

Solidifying the concepts discussed in class through practice is necessary to build your skills.

Read section 3.2.3.
Watch video Using L’Hopital’s Rule with Infinity (8:45).
Ask Copilot “How can I use L’Hopital’s rule to compute the value of a limit that is in the form 0$\wedge$0?”
Read section 3.2.4 (summary).
Do Exercises 3-4 in section 3.2.5.
Do these problems.
1. Compute each of the following limits exactly.
  1. $\displaystyle \displaystyle \lim_{x \rightarrow 0}\frac{\sin x}{x}$
  2. $\displaystyle \displaystyle \lim_{x \rightarrow 0}\frac{x}{e^{x}}$
  3. $\displaystyle \displaystyle \lim_{t \rightarrow 0}\frac{e^{t} - 1 - t}{t^{2}}$
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2. Find the sign of $\displaystyle \lim_{x \rightarrow a}\frac{f(x)}{g(x)}$ from Figure 78. Assume $f''(a) \neq 0$ and $g''(a) \neq 0\text{.}$
  
  $A coordinate graph showing two curves labeled f(x) and g(x). The blue curve f(x) lies above the x-axis near the middle and has a shallow minimum at a point above x equals a, then rises to the left and right. The red curve g(x) lies below the x-axis and has a maximum at the point directly below x equals a. Both curves touch the x-axis at the same point labeled a. The x- and y-axes are shown with arrows.$
  
  Figure 78. The graph of $f(x)$ and $g(x)\text{.}$
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Do Exercises 5-14 from section 3.2.5.
Start working on the MOMwork (MyOpenMath) assignment for this section.

Section Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Understand the development of L’Hôpital’s Rule and how it relies upon the tangent line approximation of two functions.
Use L’Hôpital’s Rule to calculate limits that have an indeterminate form “0/0”.
Use L’Hôpital’s Rule to calculate limits that have an indeterminate form “$\infty/\infty$”.
Use limits to identify locate horizontal and vertical asymptotes.
Find limits involving indeterminate products (“$0\cdot \infty$”): rewrite the limit in such a way that L’Hôpital’s Rule can be used.
Find limits involving indeterminate powers (“$\infty^{0}$”, ”$1^{\infty}$”, ”$0^{0}$”): use the identity $u=e^{\ln u}$ to rewrite the expression and find the limit of the exponent (you may get an indeterminate power and L’Hôpital’s Rule might be required at this point).
Realize which forms of limits are not indeterminate, i.e. which always result in a clear answer (e.g. “$0^{\infty}$”=0, “$*/\infty$”=0, “$*/0^{+}$”=$\infty$)

Section Additional suggestions

Watch video L’Hopital’s Rule - A Graphical Approach (10:52).
Watch video Student Problem Solving - L’Hopital’s Rule (3:20).
Watch video Limit of Quotients (6:56).
Watch video Indeterminant Products (9:07).
Watch video Horizontal Asymptotes and Limits at Infinity (6:16).
Watch video L’Hopital’s Rule and Limits of the Form 0 x Infinity (9:09).
Do these problems.
1. Evaluate $\displaystyle \lim_{x \rightarrow 1}\left( \frac{x}{x-1}- \frac{1}{\ln x}\right)\text{.}$
2. Use the graphs of $f$ and $g$ and their tangent lines at $(2,0)$ shown in Figure 79 to find $\displaystyle \lim_{x \rightarrow 2}\frac{f(x)}{g(x)}\text{.}$
  
  $A coordinate graph showing a curve labeled f and a second curve labeled g, both passing through the point (2, 0). At that point, two dotted straight lines are shown. A blue dotted line labeled y equals 1.8 times (x minus 2) has a steeper positive slope. A red dotted line labeled y equals four‑fifths times (x minus 2) has a gentler positive slope. The curve g lies above the curve f to the right of x equals 2 and below it to the left. The x‑ and y‑axes are labeled, with a grid and the point (2, 0) marked.$
  
  Figure 79. The graphs of $f(x)$ and $g(x)$ and their tangent lines at $(2,0)\text{.}$
3. Calculate $\displaystyle \lim_{x \rightarrow \infty}\left( 1 + \frac{1}{x}\right)^{x}\text{.}$ Hint: Note that this limit has the form $1^{\infty}\text{.}$ So write
  
  \begin{equation*} y=\left( 1 + \frac{1}{x}\right)^{x} \end{equation*}
  
  and find the limit of $\ln y\text{.}$
4. Put the following limits in a form that can be evaluated using L’Hôpital’s rule.
  1. $\displaystyle \lim_{x \rightarrow \infty}x^{1/x}$ Hint: Use a logarithm.
  2. $\displaystyle \lim_{x \rightarrow 0^+}x^{x}$ Hint: Use a logarithm.
  3. $\displaystyle \displaystyle \lim_{x \rightarrow 0}\frac{1}{x}- \frac{1}{\sin x}$
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5. Compute each of the following limits exactly.
  1. $\displaystyle \displaystyle \lim_{x \rightarrow \infty}\frac{\ln x}{x}$
  2. $\displaystyle \displaystyle \lim_{x \rightarrow \infty}\frac{\ln (\ln x)}{\sqrt{x}}$
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6. Calculate $\displaystyle \lim_{x \rightarrow \infty}xe^{-x}\text{.}$ Hint: Note that this limit has the form $\infty \cdot 0\text{.}$ Rewrite $\displaystyle xe^{-x}= \frac{x}{e^{x}}$ so that the limit takes the form $\displaystyle \frac{\infty}{\infty}\text{.}$
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Watch video Indeterminant Forms (6:47).
Watch video More Indeterminant Forms (6:01).