Calculus: Graphical, Numerical, Algebraic

Ross L. Finney, Franklin D. Demana, Bet K. Waits, Daniel Kennedy

Chapter 2

Limits and Continuity - all with Video Answers

Educators

Section 1

Rates of Change and Limits

Problem 1

In Exercises $1 - 4 ,$ an object dropped from rest from the top of a tall building falls $y = 16 t ^ { 2 }$ feet in the first $t$ seconds.

Find the average speed during the first 3 seconds of fall.

Alex Johnson

Alex Johnson

Numerade Educator

Problem 2

In Exercises $1 - 4 ,$ an object dropped from rest from the top of a tall building falls $y = 16 t ^ { 2 }$ feet in the first $t$ seconds.
Find the average speed during the first 4 seconds of fall.

Sooyeon Kim

Numerade Educator

Problem 3

In Exercises $1 - 4 ,$ an object dropped from rest from the top of a tall building falls $y = 16 t ^ { 2 }$ feet in the first $t$ seconds.
Find the speed of the object at $t = 3$ seconds and confirm your answer algebraically.

Alex Johnson

Numerade Educator

Problem 4

In Exercises $1 - 4 ,$ an object dropped from rest from the top of a tall building falls $y = 16 t ^ { 2 }$ feet in the first $t$ seconds.
Find the speed of the object at $t = 4$ seconds and confirm your answer algebraically.

Sooyeon Kim

Numerade Educator

Problem 5

In Exercises 5 and $6 ,$ use $\lim _ { x \rightarrow c } k = k , \lim _ { x \rightarrow c } x = c ,$ and the properties of limits to find the limit.
$$\lim _ { x \rightarrow c } \left( 2 x ^ { 3 } - 3 x ^ { 2 } + x - 1 \right)$$

Alex Johnson

Numerade Educator

Problem 6

In Exercises 5 and $6 ,$ use $\lim _ { x \rightarrow c } k = k , \lim _ { x \rightarrow c } x = c ,$ and the properties of limits to find the limit.
$$\lim _ { x \rightarrow c } \frac { x ^ { 4 } - x ^ { 3 } + 1 } { x ^ { 2 } + 9 }$$

Sooyeon Kim

Numerade Educator

Problem 7

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { x \rightarrow - 1 / 2 } 3 x ^ { 2 } ( 2 x - 1 )$$

Alex Johnson

Numerade Educator

Problem 8

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { x \rightarrow - 4 } ( x + 3 ) ^ { 1998 }$$

Sooyeon Kim

Numerade Educator

Problem 9

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { x \rightarrow 1 } \left( x ^ { 3 } + 3 x ^ { 2 } - 2 x - 17 \right)$$

Alex Johnson

Numerade Educator

Problem 10

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { y \rightarrow 2 } \frac { y ^ { 2 } + 5 y + 6 } { y + 2 }$$

Sooyeon Kim

Numerade Educator

Problem 11

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { y \rightarrow - 3 } \frac { y ^ { 2 } + 4 y + 3 } { y ^ { 2 } - 3 }$$

Alex Johnson

Numerade Educator

Problem 12

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { x \rightarrow 1 / 2 } int x$$

Sooyeon Kim

Numerade Educator

Problem 13

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { x \rightarrow - 2 } ( x - 6 ) ^ { 2 / 3 }$$

Alex Johnson

Numerade Educator

Problem 14

In Exercises $7 - 14 ,$ determine the limit by substitution. Support graphically.
$$\lim _ { x \rightarrow 2 } \sqrt { x + 3 }$$

Sooyeon Kim

Numerade Educator

Problem 15

In Exercises $15 - 18$ , explain why you cannot use substitution to determine the limit. Find the limit if it exists.
$$\lim _ { x \rightarrow - 2 } \sqrt { x - 2 }$$

Alex Johnson

Numerade Educator

Problem 16

In Exercises $15 - 18$ , explain why you cannot use substitution to determine the limit. Find the limit if it exists.
$$\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 2 } }$$

Sooyeon Kim

Numerade Educator

Problem 17

In Exercises $15 - 18$ , explain why you cannot use substitution to determine the limit. Find the limit if it exists.
$$\lim _ { x \rightarrow 0 } \frac { | x | } { x }$$

Alex Johnson

Numerade Educator

Problem 18

In Exercises $15 - 18$ , explain why you cannot use substitution to determine the limit. Find the limit if it exists.
$$\lim _ { x \rightarrow 0 } \frac { ( 4 + x ) ^ { 2 } - 16 } { x }$$

Sooyeon Kim

Numerade Educator

Problem 19

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 1 } \frac { x - 1 } { x ^ { 2 } - 1 }$$

Alex Johnson

Numerade Educator

Problem 20

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { t \rightarrow 2 } \frac { t ^ { 2 } - 3 t + 2 } { t ^ { 2 } - 4 }$$

Sooyeon Kim

Numerade Educator

Problem 21

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { 5 x ^ { 3 } + 8 x ^ { 2 } } { 3 x ^ { 4 } - 16 x ^ { 2 } }$$

Alex Johnson

Numerade Educator

Problem 22

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { \frac { 1 } { 2 + x } - \frac { 1 } { 2 } } { x }$$

Sooyeon Kim

Numerade Educator

Problem 23

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { ( 2 + x ) ^ { 3 } - 8 } { x }$$

Alex Johnson

Numerade Educator

Problem 24

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { \sin 2 x } { x }$$

Sooyeon Kim

Numerade Educator

Problem 25

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { \sin x } { 2 x ^ { 2 } - x }$$

Alex Johnson

Numerade Educator

Problem 26

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { x + \sin x } { x }$$

Sooyeon Kim

Numerade Educator

Problem 27

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { \sin ^ { 2 } x } { x }$$

Alex Johnson

Numerade Educator

Problem 28

In Exercises $19 - 28 ,$ determine the limit graphically. Confirm algebraically.
$$\lim _ { x \rightarrow 0 } \frac { 3 \sin 4 x } { \sin 3 x }$$

Sooyeon Kim

Numerade Educator

Problem 29

In Exercises 29 and 30 , use a graph to show that the limit does not exist.
$$\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 4 } { x - 1 }$$

Alex Johnson

Numerade Educator

Problem 30

In Exercises 29 and 30 , use a graph to show that the limit does not exist.
$$\lim _ { x \rightarrow 2 } \frac { x + 1 } { x ^ { 2 } - 4 }$$

Sooyeon Kim

Numerade Educator

Problem 31

In Exercises $31 - 36 ,$ determine the limit.
$$\lim _ { x \rightarrow 0 ^ { + } } \operatorname { int } x$$

Alex Johnson

Numerade Educator

Problem 32

In Exercises $31 - 36 ,$ determine the limit.
$$\lim _ { x \rightarrow 0 ^ { - } } \operatorname { int } x$$

Sooyeon Kim

Numerade Educator

Problem 33

In Exercises $31 - 36 ,$ determine the limit.
$$\lim _ { x \rightarrow 0.01 } \operatorname { int } x$$

Alex Johnson

Numerade Educator

Problem 34

In Exercises $31 - 36 ,$ determine the limit.
$$\lim _ { x \rightarrow 2 ^ { - } } \operatorname { int } x$$

Sooyeon Kim

Numerade Educator

Problem 35

In Exercises $31 - 36 ,$ determine the limit.
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { x } { | x | }$$

Alex Johnson

Numerade Educator

Problem 36

In Exercises $31 - 36 ,$ determine the limit.
$$\lim _ { x \rightarrow 0 ^ { - } } \frac { x } { | x | }$$

Sooyeon Kim

Numerade Educator

Problem 37

In Exercises 37 and $38 ,$ which of the statements are true about the
function $y = f ( x )$ graphed there, and which are false?

(a) $$\lim _ { x \rightarrow - 1 ^ { + } } f ( x ) = 1$$ (b) $$\lim _ { x \rightarrow 0 ^ { - } } f ( x ) = 0$$ (c) $$\lim _ { x \rightarrow 0 ^ { - } } f ( x ) = 1$$(d) $$\lim _ { x \rightarrow 0 ^ { - } } f ( x ) = \lim _ { x \rightarrow 0 ^ { + } } f ( x )$$(e) $$\lim _ { x \rightarrow 0 } f ( x )$$ exists (f) $$\lim _ { x \rightarrow 0 } f ( x ) = 0$$ $( \mathbf { g } ) \lim _ { x \rightarrow 0 } f ( x ) = 1$ (h) $$\lim _ { x \rightarrow 1 } f ( x ) = 1$$ (i) $$\lim _ { x \rightarrow 1 } f ( x ) = 0$$ (j) $$\lim _ { x \rightarrow 2 ^ { - } } f ( x ) = 2$$

Alex Johnson

Numerade Educator

Problem 38

In Exercises 37 and $38 ,$ which of the statements are true about the
function $y = f ( x )$ graphed there, and which are false?
(a) $$\lim _ { x \rightarrow - 1 ^ { + } } f ( x ) = 1$$ (b) $$\lim _ { x \rightarrow 2 } f ( x )$$ does not exist.
(c) $$\lim _ { x \rightarrow 2 } f ( x ) = 2$$ (d) $$\lim _ { x \rightarrow 1 ^ { - } } f ( x ) = 2$$
(e) $$\lim _ { x \rightarrow 1 ^ { + } } f ( x ) = 1$$ $$( \mathbf { f } ) \lim _ { x \rightarrow 1 } f ( x )$$ does not exist. (g) $$\lim _ { x \rightarrow 0 ^ { + } } f ( x ) = \lim _ { x \rightarrow 0 ^ { - } } f ( x )$$ (h) $\lim _ { x \rightarrow c } f ( x )$ exists at every $c$ in $( - 1,1 )$ (i) $\lim _ { x \rightarrow c } f ( x )$ exists at every $c$ in $( 1,3 )$

Sooyeon Kim

Numerade Educator

Problem 39

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathbf { a } ) \lim _ { x \rightarrow 3 ^ { - } } f ( x )$$

$$( \mathbf { b } ) \lim _ { x \rightarrow 3 ^ { + } } f ( x )$$

$$( \mathbf { c } ) \lim _ { x \rightarrow 3 } f ( x )$$

$$( \mathbf { d } ) f ( 3 )$$

Alex Johnson

Numerade Educator

Problem 40

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathbf { a } ) \lim _ { t \rightarrow - 4 ^ { - } } g ( t )$$

$$( \mathbf { b } ) \lim _ { t \rightarrow - 4 ^ { + } } g ( t )$$

$$( \mathbf { c } ) \lim _ { t \rightarrow - 4 } g ( t )$$

$$( \mathbf { c } ) g ( - 4 )$$

Sooyeon Kim

Numerade Educator

Problem 41

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathbf { a } ) \lim _ { h \rightarrow 0 ^ { - } } f ( h )$$

$$( \mathbf { b } ) \lim _ { h \rightarrow 0 ^ { + } } f ( h )$$

$$( \mathbf { c } ) \lim _ { h \rightarrow 0 } f ( h )$$

$$ ( \mathbf { d } ) f ( 0 )$$

Alex Johnson

Numerade Educator

Problem 42

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathrm { a } ) \lim _ { s \rightarrow - 2 ^ { - } } p ( s )$$

$$( \mathrm { b } ) \lim _ { s \rightarrow - 2 ^ { + } } p ( s )$$

$$( \mathrm { c } ) \lim _ { s \rightarrow - 2 } p ( s )$$

$$( \mathrm { d } ) p ( - 2 )$$

Sooyeon Kim

Numerade Educator

Problem 43

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathbf { a } ) \lim _ { x \rightarrow 0 ^ { - } } F ( x )$$

$$( \mathbf { b } ) \lim _ { x \rightarrow 0 ^ { + } } F ( x )$$

$$( \mathbf { c } ) \lim _ { x \rightarrow 0 } F ( x )$$

$$( \mathbf { d } ) F ( 0 )$$

Alex Johnson

Numerade Educator

Problem 44

In Exercises $39 - 44 ,$ use the graph to estimate the limits and value of the function, or explain why the limits do not exist.

$$( \mathrm { a } ) \lim _ { x \rightarrow 2 ^ { - } } G ( x )$$

$$( \mathrm { b } ) \lim _ { x \rightarrow 2 ^ { + } } G ( x )$$

$$( \mathrm { c } ) \lim _ { x \rightarrow 2 } G ( x )$$

$$( \mathrm { d } ) \mathrm { G } ( 2 )$$

Sooyeon Kim

Numerade Educator

Problem 45

In Exercises $45 - 48 ,$ match the function with the table.
$$y _ { 1 } = \frac { x ^ { 2 } + x - 2 } { x - 1 }$$

Alex Johnson

Numerade Educator

Problem 46

In Exercises $45 - 48 ,$ match the function with the table.
$$y _ { 1 } = \frac { x ^ { 2 } - x - 2 } { x - 1 }$$

Sooyeon Kim

Numerade Educator

Problem 47

In Exercises $45 - 48 ,$ match the function with the table.
$$y _ { 1 } = \frac { x ^ { 2 } - 2 x + 1 } { x - 1 }$$

Alex Johnson

Numerade Educator

Problem 48

In Exercises $45 - 48 ,$ match the function with the table.
$$y _ { 1 } = \frac { x ^ { 2 } + x - 2 } { x + 1 }$$

Sooyeon Kim

Numerade Educator

Problem 49

In Exercises 49 and 50 , determine the limit.

Assume that $$\lim _ { x \rightarrow 4 } f ( x ) = 0$$ and $$\lim _ { x \rightarrow 4 } g ( x ) = 3$$

(a) $$\lim _ { x \rightarrow 4 } ( g ( x ) + 3 )$$
(b) $$\lim _ { x \rightarrow 4 } x f ( x )$$
(c) $$\lim _ { x \rightarrow 4 } g ^ { 2 } ( x ) \quad \quad$$
(d) $$\lim _ { x \rightarrow 4 } \frac { g ( x ) } { f ( x ) - 1 }$$

Alex Johnson

Numerade Educator

Problem 50

In Exercises 49 and 50 , determine the limit.
Assume that $$\lim _ { x \rightarrow b } f ( x ) = 7$$ and $$\lim _ { x \rightarrow b } g ( x ) = - 3$$

(a) $$\lim _ { x \rightarrow b } ( f ( x ) + g ( x ) ) \quad \quad$$
(b) $$\lim _ { x \rightarrow b } ( f ( x ) \cdot g ( x ) )$$
(c) $$\lim _ { x \rightarrow b } 4 g ( x ) - \quad$$
(d) $$\lim _ { x \rightarrow b } \frac { f ( x ) } { g ( x ) }$$

Sooyeon Kim

Numerade Educator

Problem 51

In Exercises $51 - 54 ,$ complete parts $( a ) , ($ b) $,$ and $( c )$ for the piecewise-defined function.

(a) Draw the graph of $f .$
(b) Determine $\lim _ { x \rightarrow c ^ { + } } f ( x )$ and $\lim _ { x \rightarrow c ^ { - } } f ( x )$
(c) Writing to Learn Does $\lim _ { x \rightarrow c } f ( x )$ exist? If so, what is it? If not, explain.

$$c = 2 , f ( x ) = \left\{ \begin{array} { l l } { 3 - x , } & { x < 2 } \\ { \frac { x } { 2 } + 1 , } & { x > 2 } \end{array} \right.$$

Alex Johnson

Numerade Educator

Problem 52

In Exercises $51 - 54 ,$ complete parts $( a ) , ($ b) $,$ and $( c )$ for the piecewise-defined function.

$$c = 2 , f ( x ) = \left\{ \begin{array} { l l } { 3 - x , } & { x < 2 } \\ { 2 , } & { x = 2 } \\ { x / 2 , } & { x > 2 } \end{array} \right.$$

Sooyeon Kim

Numerade Educator

Problem 53

In Exercises $51 - 54 ,$ complete parts $( a ) , ($ b) $,$ and $( c )$ for the piecewise-defined function.

$$c = 1 , f ( x ) = \left\{ \begin{array} { l l } { \frac { 1 } { x - 1 } , } & { x < 1 } \\ { x ^ { 3 } - 2 x + 5 , } & { x \geq 1 } \end{array} \right.$$

Mutahar Mehkri

Numerade Educator

Problem 54

In Exercises $51 - 54 ,$ complete parts $( a ) , ($ b) $,$ and $( c )$ for the piecewise-defined function.

$$c = - 1 , f ( x ) = \left\{ \begin{array} { l l } { 1 - x ^ { 2 } , } & { x \neq - 1 } \\ { 2 , } & { x = - 1 } \end{array} \right.$$

Sooyeon Kim

Numerade Educator

Problem 55

In Exercises $55 - 58 ,$ complete parts $( a ) - ( d )$ for the piecewise-definedfunction. $\quad ($ a) Draw the graph of $f$ .
(b) At what points $c$ in the domain of $f$ does $\lim _ { x \rightarrow c } f ( x )$ exist?
(c) At what points $c$ does only the left-hand limit exist?
(d) At what points $c$ does only the right-hand limit exist?

$$f ( x ) = \left\{ \begin{array} { l l } { \sin x , } & { - 2 \pi \leq x < 0 } \\ { \cos x , } & { 0 \leq x \leq 2 \pi } \end{array} \right.$$

Sooyeon Kim

Numerade Educator

Problem 56

In Exercises $55 - 58 ,$ complete parts $( a ) - ( d )$ for the piecewise-definedfunction. $\quad ($ a) Draw the graph of $f$ .
(b) At what points $c$ in the domain of $f$ does $\lim _ { x \rightarrow c } f ( x )$ exist?
(c) At what points $c$ does only the left-hand limit exist?
(d) At what points $c$ does only the right-hand limit exist?
$$f ( x ) = \left\{ \begin{array} { l l } { \cos x , } & { - \pi \leq x < 0 } \\ { \sec x , } & { 0 \leq x \leq \pi } \end{array} \right.$$

Sooyeon Kim

Numerade Educator

Problem 57

In Exercises $55 - 58 ,$ complete parts $( a ) - ( d )$ for the piecewise-definedfunction. $\quad ($ a) Draw the graph of $f$ .
(b) At what points $c$ in the domain of $f$ does $\lim _ { x \rightarrow c } f ( x )$ exist?
(c) At what points $c$ does only the left-hand limit exist?
(d) At what points $c$ does only the right-hand limit exist?
$$f ( x ) = \left\{ \begin{array} { l l } { \sqrt { 1 - x ^ { 2 } } , } & { 0 \leq x < 1 } \\ { 1 , } & { 1 \leq x < 2 } \\ { 2 , } & { x = 2 } \end{array} \right.$$

Sooyeon Kim

Numerade Educator

Problem 58

In Exercises $55 - 58 ,$ complete parts $( a ) - ( d )$ for the piecewise-definedfunction. $\quad ($ a) Draw the graph of $f$ .
(b) At what points $c$ in the domain of $f$ does $\lim _ { x \rightarrow c } f ( x )$ exist?
(c) At what points $c$ does only the left-hand limit exist?
(d) At what points $c$ does only the right-hand limit exist?
$$f ( x ) = \left\{ \begin{array} { l l } { x , } & { - 1 \leq x < 0 , \text { or } 0 < x \leq 1 } \\ { 1 , } & { x = 0 } \\ { 0 , } & { x < - 1 , \text { or } x > 1 } \end{array} \right.$$

Sooyeon Kim

Numerade Educator

Problem 59

In Exercises $59 - 62 ,$ find the limit graphically. Use the Sandwich Theorem to confirm your answer.
$$\lim _ { x \rightarrow 0 } x \sin x$$

Alex Johnson

Numerade Educator

Problem 60

In Exercises $59 - 62 ,$ find the limit graphically. Use the Sandwich Theorem to confirm your answer.
$$\lim _ { x \rightarrow 0 } x ^ { 2 } \sin x$$

Sooyeon Kim

Numerade Educator

Problem 61

In Exercises $59 - 62 ,$ find the limit graphically. Use the Sandwich Theorem to confirm your answer.
$$\lim _ { x \rightarrow 0 } x ^ { 2 } \sin \frac { 1 } { x ^ { 2 } }$$

Alex Johnson

Numerade Educator

Problem 62

In Exercises $59 - 62 ,$ find the limit graphically. Use the Sandwich Theorem to confirm your answer.
$$\lim _ { x \rightarrow 0 } x ^ { 2 } \cos \frac { 1 } { x ^ { 2 } }$$

Sooyeon Kim

Numerade Educator

Problem 63

Free Fall A water balloon dropped from a window high above
the ground falls $y = 4.9 t ^ { 2 } \mathrm { m }$ in $t$ sec. Find the balloon's
(a) average speed during the first 3 sec of fall. $
(b) speed at the instant $t = 3$

Alex Johnson

Numerade Educator

Problem 64

Free Fall on a Small Airless Planet A rock released from rest to fall on a small airless planet falls $y = g t ^ { 2 } \mathrm { m }$ in $t \mathrm { sec } , g$ a constant. Suppose that the rock falls to the bottom of a crevasse 20$\mathrm { m }$ below and reaches the bottom in 4$\mathrm { sec. }$

(a) Find the value of $g .$
(b) Find the average speed for the fall.
(c) With what speed did the rock hit the bottom?

Sooyeon Kim

Numerade Educator

Problem 65

True or False If lim $f ( x ) = 2$ and $\lim f ( x ) = 2 ,$ then $$\lim _ { x \rightarrow c } f ( x ) = 2 .$$ Justify your answer.

Alex Johnson

Numerade Educator

Problem 66

True or False $\lim _ { x \rightarrow 0 } \frac { x + \sin x } { x } = 2 .$ Justify your answer.

Sooyeon Kim

Numerade Educator

Problem 67

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $\lim _ { x \rightarrow 1 ^ { - } } f ( x ) ?$

$\begin{array} { l l l l l } { \text { (A) } 5 / 2 } & { \text { (B) } 3 / 2 } & { \text { (C) } 1 } & { \text { (D) } 0 } & { \text { (E) does not exist } } \end{array}$

Alex Johnson

Numerade Educator

Problem 68

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $\lim _ { x \rightarrow 1 } + f ( x ) ?$

(A) 5$/ 2$ $( \mathrm { B } ) 3 / 2$ $( \mathbf { C } ) 1$ $( \mathbf { D } ) 0$ (E) does not exist

Sooyeon Kim

Numerade Educator

Problem 69

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $\lim _ { x \rightarrow 1 } f ( x ) ?$

$( \mathrm { A } ) 5 / 2 \quad ( \mathrm { B } ) 3 / 2$ (C) 1 (D) 0 (E) does not exist

Alex Johnson

Numerade Educator

Problem 70

In Exercises $67 - 70$ , use the following function.

$f ( x ) = \left\{ \begin{array} { l l } { 2 - x , } & { x \leq 1 } \\ { \frac { x } { 2 } + 1 , } & { x > 1 } \end{array} \right.$

Multiple Choice What is the value of $f ( 1 ) ?$

(A) 5$/ 2 \quad$ (B) 3$/ 2 \quad$ (C) (D) 0 1(E) does not exist

Sooyeon Kim

Numerade Educator

Problem 71

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.

$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$

$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$

$$f ( x ) = x \sin \frac { 1 } { x }$$

Alex Johnson

Numerade Educator

Problem 72

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = \sin \frac { 1 } { x }$$

Sooyeon Kim

Numerade Educator

Problem 73

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = \frac { 10 ^ { x } - 1 } { x }$$

Alex Johnson

Numerade Educator

Problem 74

In Exercises $71 - 74 ,$ complete the following tables and state what you believe $\lim _ { x \rightarrow 0 } f ( x )$ to be.
$$\begin{array} { c | c c c c c } { x } & { - 0.1 } & { - 0.01 } & { - 0.001 } & { - 0.0001 } & { \dots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \end{array}$$
$$\begin{array} { c c c c c } { \text { (b) } } & { 0.1 } & { 0.01 } & { 0.001 } & { 0.0001 } & { \ldots } \\ \hline f ( x ) & { ? } & { ? } & { ? } & { ? } \\ \hline \end{array}$$
$$f ( x ) = x \sin ( \ln | x | )$$

Sooyeon Kim

Numerade Educator

Problem 75

Group Activity To prove that $\lim _ { \theta \rightarrow 0 } ( \sin \theta ) / \theta = 1$ when $\theta$ is
neasured in radians, the plan is to show that the right- and left-hand limits are both $1 .$

(a) To show that the right-hand limit is $1 ,$ explain why we can restrict our attention to $0 < \theta < \pi / 2$
(b) Use the figure to show that area of $\triangle O A P = \frac { 1 } { 2 } \sin \theta$ area of sector $O A P = \frac { \theta } { 2 }$ area of $\triangle O A T = \frac { 1 } { 2 } \tan \theta$
(c)Use part $( \mathrm { b } )$ and the figure to show that for $0 < \theta < \pi / 2$ $\frac { 1 } { 2 } \sin \theta < \frac { 1 } { 2 } \theta < \frac { 1 } { 2 } \tan \theta$
(d) Show that for $0 < \theta < \pi / 2$ the inequality of part (c) can be written in the form $1 < \frac { \theta } { \sin \theta } < \frac { 1 } { \cos \theta }$
(e) Show that for $0 < \theta < \pi / 2$ the inequality of part (d) can be $\cos \theta < \frac { \sin \theta } { \theta } < 1$
(f) Use the Sandwich Theorem to show that $\lim _ { \theta \rightarrow 0 ^ { + } } \frac { \sin \theta } { \theta } = 1$
(g) Show that $( \sin \theta ) / \theta$ is an even function.
(h) Use part (g) to show that $\quad \lim _ { \theta \rightarrow 0 ^ { - } } \frac { \sin \theta } { \theta } = 1$
(i) Finally, show that $\quad \lim _ { \theta \rightarrow 0 } \frac { \sin \theta } { \theta } = 1$

Carson Merrill

Numerade Educator

Problem 76

Controlling Outputs Let $f ( x ) = \sqrt { 3 x - 2 }$
(a) Show that $\lim _ { x \rightarrow 2 } f ( x ) = 2 = f ( 2 )$
(b) Use a graph to estimate values for $a$ and $b$ so that $$1.8 < f ( x ) < 2.2$$ provided $$a < x < b$$
(c) Use a graph to estimate values for $a$ and $b$ so that $$1.99 < f ( x ) < 2.01$$ provided $$a < x < b$$

Sooyeon Kim

Numerade Educator

Problem 77

Controlling Outputs Let $f ( x ) = \sin x$
(a) Find $$f ( \pi / 6 )$$
(b) Use a graph to estimate an interval $( a , b )$ about $x = \pi / 6$ so that $$0.3 < f ( x ) < 0.7$$ provided $$a < x < b$$
(c) Use a graph to estimate an interval $( a , b )$ about $x = \pi / 6$ so that $$0.49 < f ( x ) < 0.51$$ provided $$a < x < b$$

Carson Merrill

Numerade Educator

Problem 78

Limits and Geometry Let $P \left( a , a ^ { 2 } \right)$ be a point on the parabola $y = x ^ { 2 } , a > 0 .$ Let $O$ be the origin and $( 0 , b )$ the $y$ -intercept of the perpendicular bisector of line segment $O P .$ Find $\lim _ { P \rightarrow O } b$

Sooyeon Kim

Numerade Educator