Limits

Use the Extreme Value Theorem to show that each function f in Exercises 49–54 has both a maximum and a minimum value on $[a,b]$. Then use a graphing utility to approximate values M and m in $[a,b]$ at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=3-2x^2+x^3; \\ [a,b]=[-1,1] \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f , interval [a, b], and value K in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=5-x^4; \\ [a,b]=[0,2]; \\ K=0 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f , interval [a, b], and value K in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=5-x^4; \\ [a,b]=[-2,-1]; \\ K=0 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f , interval [a, b], and value K in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=x^3-3x^2-2; \\ [a,b]=[-2,4]; \\ K=-4 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f , interval [a, b], and value K in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=x^3-3x^2-2; \\ [a,b]=[0,2]; \\ K=-4 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f , interval [a, b], and value K in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=x^3-3x^2-2; \\ [a,b]=[2,4]; \\ K=-4 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f , interval [a, b], and value K in Exercises 55– 60, there is some c∈(a, b) for which f(c) = K. Then use a graphing utility to approximate all such values c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=2+x+x^3; \\ [a,b]=[-1,2]; \\ K=3 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=x^3+2; \\ K=-15 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=-2x^2+4; \\ K=0 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\sin x; \\ K=\frac{1}{2} \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\sin x; \\ K=\frac{\sqrt{3}}{2} \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\left|3x+1\right|; \\ K=1 \end{gathered}$$

Use the Intermediate Value Theorem to show that for each function f and value K in Exercises 61–66, there must be some c∈R for which f(c) = K. You will have to select an appropriate interval [a, b] to work with. Then find or approximate one such value of c. You may assume that these functions are continuous everywhere.

$$\begin{gathered} f\left(x\right)=\left|2-3x\right|; \\ K=2 \end{gathered}$$

Find the intervals on which each function in Exercises 67–74 is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=2+5x+2x^2$

Find the intervals on which each function in Exercises 67–74 is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.
$f\left(x\right)=x^3-2x^2-3x$

Find the intervals on which each function in Exercises $67-74$ is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=\frac{x^2-4}{x^2-1}$

Find the intervals on which each function in Exercises $67-74$ is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined. 

$f\left(x\right)=\frac{\left(x+4\right)\left(x-1\right)}{2x+3}$

Find the intervals on which each function in Exercises $67-74$ is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=\left\{\begin{array}{l}x-4, if x\le 1\\ x^2-4, if x>1\end{array}\right.$

Find the intervals on which each function in Exercises $67-74$ is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=\left\{\begin{array}{l}3x+1, if x<0\\ x, if x\ge 0\end{array}\right.$

Find the intervals on which each function in Exercises $67-74$ is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=\left\{\begin{array}{l}{x}^3, if x\le 2\\ 4x-x^3, if x>2\end{array}\right.$

Find the intervals on which each function in Exercises $67-74$ is positive or negative. Make clear how your work uses the Intermediate Value Theorem and continuity. You may assume that polynomials and their quotients are continuous on the intervals on which they are defined.

$f\left(x\right)=\left\{\begin{array}{l}{x}^2-9, if x\le -2\\ x^2+x-2, if x>-2\end{array}\right.$

Explain in practical terms what the Extreme Value Theorem says about each continuous function defined in Exercises $75-77$. Then explain in practical terms what the Intermediate Value Theorem says in each situation.

Alina hasn’t cut her hair for six years. Six years ago her hair was just $2$ inches long. Now her hair is $42$ inches long. Let $H\left(t\right)$ be the function that describes the length, in inches, of Alina’s hair $t$ years after she stopped cutting it. 

Explain in practical terms what the Extreme Value Theorem says about each continuous function defined in Exercises $75-77$. Then explain in practical terms what the Intermediate Value Theorem says in each situation. 

Linda collects rain in a bucket outside her back door. Since the first day of April she has been keeping track of how the amount of water in the bucket changes as it fills with rain and evaporates. On April $1$ the bucket was empty, and today it contains $4$ inches of water. Let $w\left(t\right)$ be the height, in inches, of rainwater in the bucket $t$ days after the first day of April.

Explain in practical terms what the Extreme Value Theorem says about each continuous function defined in Exercises $75-77$. Then explain in practical terms what the Intermediate Value Theorem says in each situation.  

The number of gallons of gas in Phil’s new station wagon $t$ days after he bought it is given by the function $g\left(t\right)$. When he purchased the station wagon one year ago, the tank had $19$ gallons of gas in it. Today he ran out of gas. 

Lars was $20$ inches tall when he was born, and six foot one when he died at age $83$. Use the Intermediate Value Theorem to show that there must have been some point in Lars’s life at which his height in inches was equal to his age in years. (Hint: Think about when the difference between his height and age is zero.) 

As a vacuum cleaner salesman, Alex earns a salary of \(8,500 a year, whether he sells any vacuum cleaners or

not. In addition, for every 30 vacuum cleaners he sells, he earns a \)1,500 commission.

(a) Construct a piecewise-defined function M(v) that describes the amount of money M that Alex will make in a year if he sells v vacuum cleaners over the course of the year. Assume he sells between 0 and 90 vacuum cleaners in a year.

(b) Check that your function makes sense by using it to calculate M(0), M(30), M(59), M(61), and M(90). Then sketch a graph of M(v) on the interval 0 ≤ v ≤ 90.

(c) The piecewise-defined function M(v) is not continuous. List all the values at which M(v) fails to be continuous, and use the definition of continuity to support your answers.

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian mode 

$f\left(x\right)=\left\{\begin{array}{r}\sin \left(\frac{1}{x}\right),\text{ if }x\ne 0\\ 0,\text{ if }x=0\end{array}\right.$

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian mode 

$f\left(x\right)=\left\{\begin{array}{r}x\sin \left(\frac{1}{x}\right),\text{ if }x\ne 0\\ 0,\text{ if }x=0\end{array}\right.$

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian mode 

$f\left(x\right)=\left\{\begin{array}{r}{x}^2\sin \left(\frac{1}{x}\right),\text{ if }x\ne 0\\ 0,\text{ if }x=0\end{array}\right.$

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near $x=0$. Does it appear that f is continuous at $x=0$? Make sure your calculator is set to radian mode  

The function $f$ is continuous at $x=c.$

One immediate application of the Intermediate Value The theorem is the method of finding roots called the Bisection Method. In this problem, you will develop this method and then use it to approximate the square root of 2.

(a) Suppose f is continuous on R and that a and b are some real numbers for which f (a) is negative and f (b) is positive. Explain why the Intermediate Value Theorem guarantees that there must be some point in (a, b) where f (x) has a root.

(b) Consider the function f (x) = x² − 2. Show that f(0) is negative and f(2) is positive. What conclusion can we draw from the Intermediate Value Theorem?

(c) We can bisect the interval (0, 2) by finding the midpoint of the interval, which in this case is x = 1. Is f (1) positive or negative? Does the Intermediate Value Theorem say anything about f (x) = x² − 2 on the interval (0, 1)? What about the interval (1, 2)?

(d) Your answer to part (b) tells you that f(x) = x² − 2 must have a root somewhere in the interval (0, 2) of length 2. Your answer to part (c) tells you that f (x) = x² − 2 must have a root in a shorter interval of length 1. Now repeat! Bisect the interval of length 1 to find an interval of length $\frac{1}{2}$ on which f(x) must have a root.

(e) Describe why this Bisection Method will in general give better and better approximations for finding a root of a given function. In this particular example, with f(x) = x² − 2, why does the Bisection Method give us an approximation for $\sqrt{2}$?

Write a delta–epsilon proof that shows that the function $f\left(x\right)=3x-5$ is continuous at $x=2$. (This exercise depends on Section 1.3.)

Write a delta–epsilon proof that shows that the function $f\left(x\right)=2x+1$ is continuous at $x=5$. (This exercise depends on Section 1.3.) 

Write a delta–epsilon proof that shows that the function $f\left(x\right)=\left|x\right|$ is continuous. You may find the following inequality useful: For any real numbers $a$ and $b$, $\left|\left|a\right|-\left|b\right|\right|\le \left|a-b\right|$. (This exercise depends on Section 1.3.) 

Use what you know about one-sided limits to prove that a function $f$ is continuous at a point $x=c$ if and only if it is both left and right continuous at $x=c$.

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x=c$.

$f\left(x\right)=\left\{\begin{array}{r}2-x,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ x^2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational },\end{array}c=2\right.$

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x=c$.

$f\left(x\right)=\left\{\begin{array}{r}2-x,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ rational }\\ x^2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }x\text{ irrational },\end{array}c=1\right.$

Write a delta–epsilon proof that proves that f is continuous on its domain. In each case, you will need to assume that δ is less than or equal to 1. 

$f\left(x\right)=x^{-1}$

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$.  

$f\left(x\right)=x^2$

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$.  

$f\left(x\right)=x^3$

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$.

$f\left(x\right)=x^{-2}$

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$.  

$f\left(x\right)=x^{-1/2}$

Read the section and make your own summary of the material.

Values of transcendental functions: Without a calculator, find each of the function values that follow. For some values the answer may be undefined. 

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

a. True or False: The limit of a difference of functions as \(x\rightarrow c\) is equal to the difference of the limits of those functions as \(x\rightarrow c\) , provided that all limits involved exist.

b. True or False: If \(f(x)\) is within \(0.25\) unit of \(7\) and \(g(x)\) is within \(0.25\) unit of \(2\), then \(f(x) + g(x)\) is within \(0.5\) unit of \(9\).

c.  True or False: If \(f(x)\) is within \(0.25\) unit of \(7\) and \(g(x)\) is within \(0.25\) unit of \(2\), then \(f(x)g(x)\) is within \(0.5\) unit of \(9\).

d. True or False: Every algebraic function \(f\) is continuous at every real number \(x = c\)

e. True or False: Every power function \(f(x) = Ax^{k}\) is continuous at the point \(x = 2\). 

f. True or False: The function \(f(x) = sec x\) is continuous at \(x=\frac{\pi }{2}\).

g. True or False: The value of \(\frac{(x-c)f(x)}{(x-c)g(x)}\) at \(x = c\) is equal to the limit of \(\frac{f(x)}{g(x)}\) at \(x=x\).

h. True or False: The limit of \(\frac{(x-c)f(x)}{(x-c)g(x)}\) as \(x\rightarrow\) is equal to the limit of \(\frac{f(x)}{g(x)}\) as \(x\rightarrow c\).

The \(\delta-\varepsilon\) definition of limit: Write each limit statement that follows in terms of the \(\delta-\varepsilon\)  definition of limit. Then approximate the largest value of \(\delta\)  corresponding to \(\varepsilon =0.5\), and illustrate this choice of \(\delta\) on a graph of \(f\).

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

1. Two limits that are initially in an indeterminate form but can be solved with the Cancellation Theorem.
2. Two limits that can be solved with the Squeeze Theorem.
3. Two limits that we do not yet know how to calculate.

State the constant multiple rule, sum rule, product rule, quotient rule, and composition rule for limits.