Prove the Function aÅ¡x is Continuous on Its Domain 0 a

Problem 1

Referring to Figure $14,$ state whether $f(x)$ is left- or right-continuous (or neither) at each point of discontinuity. Does $f(x)$ have any removable discontinuities?

Maria Dascalu

Numerade Educator

Problem 2

Exercises $2-4$ refer to the function $g(x)$ in Figure 15.
\begin{equation}\begin{array}{l}{\text { State whether } g(x) \text { is left-or right-continuous (or neither) at each }} \\ {\text { of its points of discontinuity. }}\end{array}\end{equation}

Dishary Hossain

Numerade Educator

Problem 3

Exercises $2-4$ refer to the function $g(x)$ in Figure 15.
\begin{equation}\begin{array}{l}{\text { At which point } c \text { does } g(x) \text { have a removable discontinuity? How }} \\ {\text { should } g(c) \text { be redefined to make } g \text { continuous at } x=c ?}\end{array}\end{equation}

Maria Dascalu

Numerade Educator

Problem 4

Exercises $2-4$ refer to the function $g(x)$ in Figure 15.
\begin{equation}\begin{array}{l}{\text { Find the point } c_{1} \text { at which } g(x) \text { has a jump discontinuity but is left- }} \\ {\text { continuous. How should } g\left(c_{1}\right) \text { be redefined to make } g \text { right-continuous }} \\ {\text { at } x=c_{1} ?}\end{array}\end{equation}

Maria Dascalu

Numerade Educator

Problem 5

In Figure $16,$ determine the one-sided limits at the points of discontinuity. Which discontinuity is removable and how should $f$ be redefined to make it continuous at this point?

Maria Dascalu

Numerade Educator

Problem 6

Suppose that $f(x)=2$ for $x<3$ and $f(x)=-4$ for $x>3$.
\begin{equation}\begin{array}{l}{\text { (a) What is } f(3) \text { if } f \text { is left-continuous at } x=3 ?} \\ {\text { (b) What is } f(3) \text { if } f \text { is right-continuous at } x=3 ?}\end{array}\end{equation}

Dishary Hossain

Numerade Educator

Problem 7

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=x+\sin x$$

Maria Dascalu

Numerade Educator

Problem 8

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=x \sin x$$

Dishary Hossain

Numerade Educator

Problem 9

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=3 x+4 \sin x$$

Maria Dascalu

Numerade Educator

Problem 10

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=3 x^{3}+8 x^{2}-20 x$$

Dishary Hossain

Numerade Educator

Problem 11

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=\frac{1}{x^{2}+1}$$

Maria Dascalu

Numerade Educator

Problem 12

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=\frac{x^{2}-\cos x}{3+\cos x}$$

Dishary Hossain

Numerade Educator

Problem 13

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=\cos \left(x^{2}\right)$$

Maria Dascalu

Numerade Educator

Problem 14

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=\tan ^{-1}\left(4^{x}\right)$$

Dishary Hossain

Numerade Educator

Problem 15

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=e^{x} \cos 3 x$$

Maria Dascalu

Numerade Educator

Problem 16

In Exercises $7-16,$ use the Laws of Continuity and Theorems 2 and 3 to show that the function is continuous.
$$f(x)=\ln \left(x^{4}+1\right)$$

Dishary Hossain

Numerade Educator

Problem 17

In Exercises $17-34$ , determine the points of discontinuity. State the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left-or right-continuous.
$$f(x)=\frac{1}{x}$$

Maria Dascalu

Numerade Educator

Problem 18

Dishary Hossain

Numerade Educator

Problem 19

Maria Dascalu

Numerade Educator

Problem 20

Dishary Hossain

Numerade Educator

Problem 21

Maria Dascalu

Numerade Educator

Problem 22

Dishary Hossain

Numerade Educator

Problem 23

Maria Dascalu

Numerade Educator

Problem 24

Dishary Hossain

Numerade Educator

Problem 25

Maria Dascalu

Numerade Educator

Problem 26

Dishary Hossain

Numerade Educator

Problem 27

Maria Dascalu

Numerade Educator

Problem 28

Maria Dascalu

Numerade Educator

Problem 29

Linh Vu

Numerade Educator

Problem 30

Linh Vu

Numerade Educator

Problem 31

Linh Vu

Numerade Educator

Problem 32

Linh Vu

Numerade Educator

Problem 33

Maria Dascalu

Numerade Educator

Problem 34

Dishary Hossain

Numerade Educator

Problem 35

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=2 \sin x+3 \cos x$$

Linh Vu

Numerade Educator

Problem 36

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\sqrt{x^{2}+9}$$

Dishary Hossain

Numerade Educator

Problem 37

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\sqrt{x} \sin x$$

Linh Vu

Numerade Educator

Problem 38

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\frac{x^{2}}{x+x^{1 / 4}}$$

Dishary Hossain

Numerade Educator

Problem 39

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=x^{2 / 3} 2^{x}$$

Linh Vu

Numerade Educator

Problem 40

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=x^{1 / 3}+x^{3 / 4}$$

Dishary Hossain

Numerade Educator

Problem 41

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=x^{-4 / 3}$$

Linh Vu

Numerade Educator

Problem 42

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\ln \left(9-x^{2}\right)$$

Dishary Hossain

Numerade Educator

Problem 43

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\tan ^{2} x$$

Linh Vu

Numerade Educator

Problem 44

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\cos \left(2^{x}\right)$$

Dishary Hossain

Numerade Educator

Problem 45

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=\left(x^{4}+1\right)^{3 / 2}$$

Linh Vu

Numerade Educator

Problem 46

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=e^{-x^{2}}$$

Dishary Hossain

Numerade Educator

Problem 47

Linh Vu

Numerade Educator

Problem 48

In Exercises 35-48, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.
$$f(x)=9^{\tan x}$$

Dishary Hossain

Numerade Educator

Problem 49

Show that the function
$$f(x)=\left\{\begin{array}{ll}{x^{2}+3} & {\text { for } x<1} \\ {10-x} & {\text { for } 1 \leq x \leq 2} \\ {6 x-x^{2}} & {\text { for } x>2}\end{array}\right.$$
is continuous for $x \neq 1,2$ . Then compute the right-and left-hand limits at $x=1,2,$ and determine whether $f(x)$ is left-continuous, right-continuous, or continuous at these points (Figure 17$)$ .

Suzanne W.

Numerade Educator

Problem 50

Sawtooth Function Draw the graph of $f(x)=x-[x]$ . At which points is $f$ discontinuous? Is it left-or right-continuous at those points?

Dishary Hossain

Numerade Educator

Problem 51

In Exercises $51-54,$ sketch the graph of $f(x) .$ At each point of discontinuity, state whether $f$ is left-or right-contimuous.
$$f(x)=\left\{\begin{array}{ll}{x^{2}} & {\text { for } x \leq 1} \\ {2-x} & {\text { for } x>1}\end{array}\right.$$

Linh Vu

Numerade Educator

Problem 52

In Exercises $51-54,$ sketch the graph of $f(x) .$ At each point of discontinuity, state whether $f$ is left-or right-contimuous.
$$f(x)=\left\{\begin{array}{ll}{x+1} & {\text { for } x<1} \\ {\frac{1}{x}} & {\text { for } x \geq 1}\end{array}\right.$$

Dishary Hossain

Numerade Educator

Problem 53

In Exercises $51-54,$ sketch the graph of $f(x) .$ At each point of discontinuity, state whether $f$ is left-or right-contimuous.
$$f(x)=\left\{\begin{array}{ll}{\frac{x^{2}-3 x+2}{|x-2|}} & {x \neq 2} \\ {0} & {x=2}\end{array}\right.$$

Linh Vu

Numerade Educator

Problem 54

In Exercises $51-54,$ sketch the graph of $f(x) .$ At each point of discontinuity, state whether $f$ is left-or right-contimuous.
$$f(x)=\left\{\begin{array}{ll}{x^{3}+1} & {\text { for }-\infty < x \leq 0} \\ {-x+1} & {\text { for } 0 < x < 2} \\ {-x^{2}+10 x-15} & {\text { for } x \geq 2}\end{array}\right.$$

Suzanne W.

Numerade Educator

Problem 55

Show that the function
$$f(x)=\left\{\begin{array}{ll}{\frac{x^{2}-16}{x-4}} & {x \neq 4} \\ {10} & {x=4}\end{array}\right.$$
has a removable discontinuity at $x=4$.

Linh Vu

Numerade Educator

Problem 56

GU Define $f(x)=x \sin \frac{1}{x}+2$ for $x \neq 0 .$ Plot $f(x) .$ How should $f(0)$ be defined so that $f$ is continuous at $x=0 ?$

Dishary Hossain

Numerade Educator

Problem 57

In Exercises $57-59,$ find the value of the constant $(a, b,$ or $c)$ that makes the function continuous.
$$f(x)=\left\{\begin{array}{ll}{x^{2}-c} & {\text { for } x<5} \\ {4 x+2 c} & {\text { for } x \geq 5}\end{array}\right.$$

Linh Vu

Numerade Educator

Problem 58

In Exercises $57-59,$ find the value of the constant $(a, b,$ or $c)$ that makes the function continuous.
$$f(x)=\left\{\begin{array}{ll}{2 x+9 x^{-1}} & {\text { for } x \leq 3} \\ {-4 x+c} & {\text { for } x>3}\end{array}\right.$$

Dishary Hossain

Numerade Educator

Problem 59

In Exercises $57-59,$ find the value of the constant $(a, b,$ or $c)$ that makes the function continuous.
$$f(x)=\left\{\begin{array}{ll}{x^{-1}} & {\text { for } x<-1} \\ {a x+b} & {\text { for }-1 \leq x \leq \frac{1}{2}} \\ {x^{-1}} & {\text { for } x>\frac{1}{2}}\end{array}\right.$$

Linh Vu

Numerade Educator

Problem 60

Define
$$g(x)=\left\{\begin{array}{ll}{x+3} & {\text { for } x<-1} \\ {c x} & {\text { for }-1 \leq x \leq 2} \\ {x+2} & {\text { for } x>2}\end{array}\right.$$
Find a value of $c$ such that $g(x)$ is
\begin{equation}\text { (a) left-continuous }\quad\quad\quad\quad \text { (b) right-continuous }\end{equation}
In each case, sketch the graph of $g(x)$ .

Suzanne W.

Numerade Educator

Problem 61

Define $g(t)=\tan ^{-1}\left(\frac{1}{t-1}\right)$ for $t \neq 1 .$ Answer the following questions, using a plot if necessary.
\begin{equation}\begin{array}{l}{\text { (a) Can } g(1) \text { be defined so that } g(t) \text { is continuous at } t=1 ?} \\ {\text { (b) How should } g(1) \text { be defined so that } g(t) \text { is left-continuous at } t=1 ?}\end{array}\end{equation}

Suzanne W.

Numerade Educator

Problem 62

Each of the following statements is $false$. For each statement, sketch the graph of a function that provides a counterexample.
\begin{equation}\text { a. }\lim _{x \rightarrow a} f(x) \text { exists, then } f(x) \text { is continuous at } x=a. \\ {\text { (b) If } f(x) \text { has a jump discontinuity at } x=a, \text { then } f(a) \text { is equal to }} \\ {\text { either }} \lim _{x \rightarrow a-} f(x) \text { or } \lim _{x \rightarrow a+} f(x).\end{equation}

Dishary Hossain

Numerade Educator

Problem 63

In Exercises $63-66,$ draw the graph of a function on $10,51$ with the given properties.
\begin{equation}{ f(x) \text { is not continuous at } x=1, \text { but }} \lim _{x \rightarrow 1+} f(x) \text { and } \lim _{x \rightarrow 1-} f(x) \\ {\text { exist and are equal. }}\end{equation}

Linh Vu

Numerade Educator

Problem 64

In Exercises $63-66,$ draw the graph of a function on $10,51$ with the given properties.
\begin{equation}\begin{array}{l}{ f(x) \text { is left-continuous but not continuous at } x=2 \text { and right- }} \\ {\text { continuous but not continuous at } x=3 \text { . }}\end{array}\end{equation}

Dishary Hossain

Numerade Educator

Problem 65

In Exercises $63-66,$ draw the graph of a function on $10,51$ with the given properties.
\begin{equation}\begin{array}{l}{ f(x) \text { has a removable discontinuity at } x=1, \text { a jump discontinuity }} \\ {\text { at } x=2, \text { and }}\end{array}\end{equation}
\begin{equation}\lim _{x \rightarrow 3^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow 3+} f(x)=2\end{equation}

Linh Vu

Numerade Educator

Problem 66

In Exercises $63-66,$ draw the graph of a function on $10,51$ with the given properties.
\begin{equation}\begin{array}{l}{ f(x) \text { is right- but not left-continuous at } x=1, \text { left-but not right- }} \\ {\text { continuous at } x=2, \text { and neither left-nor right-continuous at } x=3 \text { . }}\end{array}\end{equation}

Dishary Hossain

Numerade Educator

Problem 67

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow-1}\left(2 x^{3}-4\right)$$

Linh Vu

Numerade Educator

Problem 68

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 2}\left(5 x-12 x^{-2}\right)$$

Dishary Hossain

Numerade Educator

Problem 69

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 3} \frac{x+2}{x^{2}+2 x}$$

Linh Vu

Numerade Educator

Problem 70

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow \pi} \sin \left(\frac{x}{2}-\pi\right)$$

Dishary Hossain

Numerade Educator

Problem 71

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow \frac{\pi}{4}} \tan (3 x)$$

Linh Vu

Numerade Educator

Problem 72

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow \pi} \frac{1}{\cos x}$$

Dishary Hossain

Numerade Educator

Problem 73

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 4} x^{-5 / 2}$$

Linh Vu

Numerade Educator

Problem 74

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 2} \sqrt{x^{3}+4 x}$$

Dishary Hossain

Numerade Educator

Problem 75

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow-1}\left(1-8 x^{3}\right)^{3 / 2}$$

Linh Vu

Numerade Educator

Problem 76

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 2}\left(\frac{7 x+2}{4-x}\right)^{2 / 3}$$

Dishary Hossain

Numerade Educator

Problem 77

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 3} 10^{x^{2}-2 x}$$

Linh Vu

Numerade Educator

Problem 78

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow-\frac{\pi}{2}} 3^{\sin x}$$

Dishary Hossain

Numerade Educator

Problem 79

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 4} \sin ^{-1}\left(\frac{x}{4}\right)$$

Linh Vu

Numerade Educator

Problem 80

In Exercises $67-80,$ evaluate using substitution.
$$\lim _{x \rightarrow 0} \tan ^{-1}\left(e^{x}\right)$$

Dishary Hossain

Numerade Educator

Problem 81

Suppose that $f(x)$ and $g(x)$ are discontinuous at $x=c .$ Does it follow that $f(x)+g(x)$ is discontinuous at $x=c ?$ If not, give a counterexample. Does this contradict Theorem 1$(1) ?$

Linh Vu

Numerade Educator

Problem 82

Prove that $f(x)=|x|$ is continuous for all $x .$ Hint: To prove continuity at $x=0,$ consider the one-sided limits.

Dishary Hossain

Numerade Educator

Problem 83

Use the result of Exercise 82 to prove that if $g(x)$ is continuous, then $f(x)=\lg (x) |$ is also continuous.

Linh Vu

Numerade Educator

Problem 84

Which of the following quantities would be represented by continuous functions of time and which would have one or more discontinuities?
\begin{equation}\begin{array}{l}{\text { (a) Velocity of an airplane during a flight }} \\ {\text { (b) Temperature in a room under ordinary conditions }} \\ {\text { (c) Value of a bank account with interest paid yearly }} \\ {\text { (d) The salary of a teacher }} \\ {\text { (e) The population of the world }}\end{array}\end{equation}

Dishary Hossain

Numerade Educator

Problem 85

In $2009,$ the federal income tax $T(x)$ on income of $x$ dollars (up to $\$ 82,250 )$ was determined by the formula
$$T(x)=\left\{\begin{array}{ll}{0.10 x} & {\text { for } 0 \leq x<8350} \\ {0.15 x-417.50} & {\text { for } 8350 \leq x<33,950} \\ {0.25 x-3812.50} & {\text { for } 33,950 \leq x<82,250}\end{array}\right.$$
Sketch the graph of $T(x) .$ Does $T(x)$ have any discontinuities? Explain why if $T(x)$ had a jump discontinuity, it might be advantageous in some situations to earn less money.

Problem 86

If $f(x)$ has a removable discontinuity at $x=c,$ then it is possible to redefine $f(c)$ so that $f(x)$ is continuous at $x=c$ . Can this be done in more than one way?

Linh Vu

Numerade Educator

Problem 87

Give an example of functions $f(x)$ and $g(x)$ such that $f(g(x))$ is continuous but $g(x)$ has at least one discontinuity.

Linh Vu

Numerade Educator

Problem 88

Continuous at Only One Point Show that the following function is continuous only at $x=0$ :
$$f(x)=\left\{\begin{array}{ll}{x} & {\text { for } x \text { rational }} \\ {-x} & {\text { for } x \text { irrational }}\end{array}\right.$$

Linh Vu

Numerade Educator

Problem 89

Show that $f(x)$ is a discontinuous function for all $x$ where $f(x)$ is defined as follows:
$$f(x)=\left\{\begin{array}{ll}{1} & {\text { for } x \text { rational }} \\ {-1} & {\text { for } x \text { inrational }}\end{array}\right.$$
Show that $f(x)^{2}$ is continuous for all $x$.

Linh Vu

Numerade Educator

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Source: https://www.numerade.com/books/chapter/limits-2/?section=3614