Deck 1: Limits

Question 1

Find the equation of a possible function $f$ with $f ( 0 ) = 2 , \quad \lim _ { x \rightarrow 1 ^ { - } } f ( x ) = \infty ,$ and $\lim f ( x ) = \infty$

Answer 1

$f ( x ) = \frac { 2 } { ( x - 1 ) ^ { 2 } } \text { is one of many possible answers }$

Question 2

Find the equation of a possible function $f$ with $f ( 0 ) = - 2 , \quad \lim _ { x \rightarrow - 2 ^ { - } } f ( x ) = \infty ,$ and $\lim _ { x \rightarrow - 2 ^ { + } } f ( x ) = - \infty$

Answer 2

$f ( x ) = \frac { - 4 } { x + 2 } \text { is one of many possible answers }$

Question 3

Find the equation of a possible function $f$ with $\lim f ( x ) = 0$ and $\lim _ { x \rightarrow + \infty } f ( x ) = 0$ not defined at $x = 3$ , and $f ( 0 ) = - 1$

Answer 3

$f ( x ) = \frac { 3 } { x - 3 } \text { is one of many possible answers }$

Question 4

$f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 1 & \text { if } x < - 1 \\3 & \text { if } - 1 \leq x < 2 \\2 x - 1 & \text { if } x \geq 2\end{array} \right.$ Find $\lim _ { x \rightarrow - 1 } f ( x )$ if it exists.

A) 2
B) Does not exist
C) 3
D) 0

Question 5

$f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 1 & \text { if } x < - 1 \\3 & \text { if } - 1 \leq x < 2 \\2 x - 1 & \text { if } x \geq 2\end{array} \right.$ Find $\lim _ { x \rightarrow 2 } f ( x )$ if it exists.

A) Does not exist
B) 2
C) 0
D) 3

Question 6

$f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 1 & \text { if } x < - 1 \\3 & \text { if } - 1 \leq x < 2 \\2 x - 1 & \text { if } x \geq 2\end{array} \right.$ Find $\lim _ { x \rightarrow 0 } f ( x )$ if it exists.

A) 1
B) 3
C) -1
D) Does not exist

Question 7

$f ( x ) = \left\{ \begin{array} { c } x ^ { 2 } \text { if } x < 3 \\2 x + 2 \text { if } x \geq 3\end{array} \right.$ Find $\lim _ { x \rightarrow 3 } f ( x )$ if it exists. Otherwise, explain by one-sided limits.

Question 8

Does $\lim _ { x \rightarrow 2 } \frac { | x - 2 | } { x - 2 }$ exist? Explain why or why not.

Question 9

Find $\lim _ { x \rightarrow - 1 } \frac { | x + 1 | } { x + 1 }$ , if it exists.

A) -1
B) 1
C) Does not exist
D) 0

Question 10

Find $\lim _ { x \rightarrow - 2 } \frac { | x - 2 | } { x - 2 }$ , if it exists.

A) 1
B) - 1
C) Does not exist
D) 4

Question 11

Write the limit as a formal statement involving $\delta \text { and } \varepsilon$ $\lim _ { x \rightarrow 11 } \sqrt { x - 2 } = 3$

Question 12

Write the limit as a formal statement involving $\delta \text { and } \varepsilon$ $\lim _ { x \rightarrow 3 } \frac { x ^ { 2 } - 9 } { x + 3 } = 0$

Question 13

For $\lim _ { x \rightarrow 2 } x ^ { 3 } = 8$ and $\varepsilon = 0.2$ , approximate the largest value of $\delta$ that satisfies the definition of the limit.

A) 0.0011
B) 0.0012
C) 0.01
D) 0.001

Question 14

For $\lim _ { x \rightarrow 3 } x ^ { 2 } = 9$ and $\varepsilon = 0.35$ , approximate the largest value of $\delta$ that satisfies the definition of the limit.

A) 0.04
B) 0.05
C) 0.06
D) 0.03

Question 15

Find the largest possible value of $\delta$ that makes the following statement true: "If $0 < | x - 1 | < \delta$ then $| ( 2 x + 3 ) - 5 | < 0.2 "$

A) 0.2
B) 0.05
C) 0.1
D) None of these

Question 16

Which of the following statements are true?
A. If $x \neq a$ , then $| x - a | < 0$ .
B. If $| x - a | > 0$ , then $x \neq a$ .
C. $x$ is a solution of $0 < | x - a | < \delta$ if, and only if, $a - \delta < x < a + \delta$ .
D. $\lim ( 3 x - 1 ) = 5$ means that for all $\delta > 0$ , there is some $\varepsilon > 0$ such that if $0 < | x - 5 | < \hat { \delta }$ , then $\left| \left( \begin{array} { c } x \rightarrow 2 \\ 3 x - 1 \end{array} \right) - 2 \right| < \varepsilon$ .

A) Only A is true.
B) A and D are true.
C) B and C are true.
D) B and D are true.

Question 17

Given $\lim _ { x \rightarrow 2 } ( x + 3 ) = 5$ find a $\delta > 0$ in terms of an $\varepsilon > 0$ so that if $0 < | x - c | < \delta$ then $| f ( x ) - L | < \varepsilon$

A) /2
B) /3
C) 1
D) None of these.

Question 18

Given $\lim _ { x \rightarrow - 4 } ( 2 x + 6 ) = - 2$ find a $\delta > 0$ in terms of an $\varepsilon > 0$ so that if $0 < | x - c | < \delta$ then $| f ( x ) - L | < \varepsilon$

A) /3
B) /2
C) 1
D) None of these

Question 19

Given $\lim _ { x \rightarrow - 3 } ( 2 - 3 x ) = 11$ find a $\delta > 0$ in terms of an $\varepsilon > 0$ so that if $0 < | x - c | < \delta$ then $| f ( x ) - L | < \varepsilon$

A) $E$
B) $\varepsilon / 3$
C) $\varepsilon / 2$
D) None of these

Question 20

Is $f ( x ) = \sqrt { x }$ continuous at $x = - 1 ?$ Explain why or why not.

Question 21

Which of the following statements are true?
A. If $\lim f ( x )$ exists, then $f$ is continuous at $x = c$ .
B. If $f ( 2 ) = - 3$ , and $f ( 6 ) = - 8$ , then there must be a value $c$ at which $f ( c ) = - 5$ .
C. If $f$ is continuous on the closed interval [1, 4], then $f$ is continuous at every point in [1,4].
D. If $f$ is continuous on the interval $( 1,4 )$ , then $f$ must have a maximum value and a minimum value on $( 1,4 )$ .

A) A and C are true.
B) A, C, and D are true.
C) C and D are true.
D) Only C is true.

Question 22

Is $f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.$ continuous at $x = 2 ?$ Explain why or why not.

Question 23

Is $f ( x ) = \left\{ \begin{array} { c l } x - 2 & \text { if } x < 2 \\- ( x - 2 ) & \text { if } x \geq 2\end{array} \right.$ continuous at $x = 0 ?$ Explain why or why not.

Question 24

Find all the discontinuities of $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & \text { if } x \leq 3 \\9 & \text { if } 3 < x < 5 \\2 x + 1 & \text { if } x \geq 5\end{array} \right.$

A) 5 and 9
B) 5
C) 9
D) f is continuous everywhere.

Question 25

Find all the discontinuities of $f ( x ) = \left\{ \begin{array} { c c } 2 & \text { if } x \leq - 1 \\\frac { 1 } { x } & \text { if } - 1 < x < 1 \\2 x - 1 & \text { if } x \geq 1\end{array} \right.$

A) 0
B) -1
C) 0 and -1
D) f is continuous everywhere.

Question 26

Find all the discontinuities of $f ( x ) = \left\{ \begin{array} { c c } \sqrt [ 3 ] { x - 1 } & \text { if } x \leq 0 \\- 1 & \text { if } 0 < x < 3 \\2 x - 7 & \text { if } x \geq 3\end{array} \right.$

A) 0 and 3
B) 3
C) 0
D) f is continuous everywhere.

Question 27

Use the Intermediate Value Theorem to find a $c \in ( - \infty , \infty )$ for $f ( x ) = x ^ { 3 } - x ^ { 2 } - 1$ on the interval [-1, 3] such that $f ( c ) = 3 .$

Question 28

Use the Intermediate Value Theorem to find a $c \in ( - \infty , \infty )$ for $f ( x ) = x ^ { 3 } + 2$ on the interval [-2, 3] such that $f ( c ) = 1$

Question 29

Show that the function $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 2$ has at least one zero on the interval [-1, 4].

Question 30

Show that the function $f ( x ) = x ^ { 3 } + x ^ { 2 } - x + 2$ has at least one zero on the interval [-3, 0].

Question 31

Given $f ( x ) = \frac { x - 5 } { x ^ { 2 } - 25 }$ , find all the $X$ values where $f$ is not continuous.

A) - 5
B) 5
C) 0 and - 5
D) - 5 and 5

Question 32

Given $f ( x ) = \frac { 4 x } { x ^ { 2 } + 3 x }$ , find all the $X$ values where $f$ is not continuous.

A) - 3
B) 0
C) 0 and - 3
D) - 3 and 3

Question 33

Given $f ( x ) = \frac { | x - 5 | } { x - 5 }$ , find all the $X$ values where $f$ is not continuous.

Question 34

Given $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 2$ , find all the $X$ values where $f$ is not continuous.?

Question 35

Given that $f ( x ) = \left\{ \begin{array} { c } 4 x ^ { 2 } + 2 \text { if } x < 0 \\3 x + a \text { if } x \geq 0\end{array} \right.$ , find a real number $a$ a that makes $f$ continuous at $x = 0 .$

Question 36

Given $f ( x ) = \left\{ \begin{array} { c c } a x + 3 & \text { if } x < 0 \\2 & \text { if } \mathrm { x } = 0 \\\frac { 3 } { x + 1 } & \text { if } \mathrm { x } > 0\end{array} \right.$ can we find a real number $a$ that makes $f$ continuous at $x = 0 .$ ?

Question 37

Given the function $f$ below, define $f ( 2 )$ , so that $f$ is continuous at $x = 2$ , $f ( x ) = \left\{ \begin{array} { l } 2 x - 3 \text { if } x < 2 \\5 - x ^ { 2 } \text { if } x > 2\end{array} \right.$

Question 38

Find all discontinuities of $f ( x ) = \frac { 16 x - 32 } { x ^ { 2 } - 4 }$ ; determine if they are removable or non-removable.

Question 39

Given that $f ( x ) = \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } - 1 }$ , define $f ( - 1 )$ so that $f$ is continuous at $x = - 1$

A) 1/2
B) 2
C) -2
D) -1/2

Question 40

Which of the following is the equation of a function $f$ that has a vertical asymptote at $x = 2$ , a removable discontinuity at $x = - 2$ , and $f ( 0 ) = - 2$ ?

A) $f ( x ) = \frac { 3 x - 9 } { x ^ { 2 } - 4 }$
B) $f ( x ) = \frac { 4 x + 8 } { x ^ { 2 } - 4 }$
C) $f ( x ) = \frac { 2 x - 6 } { x ^ { 2 } - 4 }$
D) $f ( x ) = \frac { 4 x - 8 } { x ^ { 2 } - 4 }$

Question 41

Suppose that $f$ and $g$ are functions with $\lim _ { x \rightarrow 3 } f ( x ) = 4 , \lim _ { x \rightarrow 3 } g ( x ) = 6$ Find $\lim _ { x \rightarrow 3 } \frac { 2 - f ( x ) } { g ( x ) }$

A) 1/3
B) 3
C) -3
D) -1/3

Question 42

Suppose that $f \text { and } g$ are functions with $\lim _ { x \rightarrow 3 } f ( x ) = 4 , \lim _ { x \rightarrow 3 } g ( x ) = 6$ Find $\lim _ { x \rightarrow 3 } \sqrt [ 3 ] { f ( x ) g ( x ) + 3 }$

A) -3
B) 6
C) 3
D) 13

Question 43

Suppose that $f$ and $g$ are functions with $\lim _ { x \rightarrow - 2 } f ( x ) = - 1 , \quad \lim _ { x \rightarrow - 2 } g ( x ) = 4$ Find $\lim _ { x \rightarrow - 2 } ( 5 f ( x ) - 3 g ( x ) )$

Question 44

Suppose that $f$ and $g$ are continuous functions with $\lim _ { x \rightarrow 3 } f ( x ) = 4 , \lim _ { x \rightarrow 2 } g ( x ) = 3$ Find $\lim _ { x \rightarrow 2 } f ( g ( x ) )$ , if possible. Otherwise, explain why not.

Question 45

Suppose that $f$ and $g$ are functions with $\lim _ { x \rightarrow 3 } f ( x ) = 4 , \lim _ { x \rightarrow 2 } g ( x ) = 3$ Find $\lim _ { x \rightarrow 2 } f ( g ( x ) )$ , if possible. Otherwise, explain why not.?

Question 46

Find all the removable discontinuities of $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$

Question 47

Find all removable discontinuities of $f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { x ^ { 2 } - 5 x + 4 }$

A) 1
B) -1
C) 4
D) - 4

Question 48

Find all the vertical asymptotes of $f ( x ) = \frac { x - 2 } { x ^ { 2 } - 4 }$

Question 49

Find $\lim _ { x \rightarrow 2 } \frac { 3 + 2 x } { x + 2 }$

Question 50

Find $\lim _ { x \rightarrow - 2 } \frac { 3 + 2 x } { x + 2 }$

Question 51

Find $\lim _ { x \rightarrow - 2 } \frac { 4 x + 2 x ^ { 2 } } { x + 2 }$

Question 52

Find $\lim _ { x \rightarrow - 2 } \frac { 4 + 2 x } { x ^ { 2 } + 2 x }$

Question 53

Find $\lim _ { x \rightarrow - 2 } \frac { x ^ { 2 } + x - 2 } { x ^ { 2 } + 5 x + 6 }$

Question 54

Find $\lim _ { x \rightarrow - 3 } \frac { x } { x + 3 }$

A) - $\infty$
B) + $\infty$
C) Does not exist
D) 0

Question 55

Find $\lim _ { x \rightarrow 2 ^ { + } } \frac { x } { x - 2 }$ .

A) - $\infty$
B) + $\infty$
C) Does not exist
D) 0

Question 56

Find $\lim _ { x \rightarrow 2 ^ { - } } \frac { x } { x - 2 }$

Question 57

Find $\lim _ { x \rightarrow - 2 } \frac { 8 + 8 x + 4 x ^ { 2 } + x ^ { 3 } } { x ^ { 2 } + x - 2 }$

A) -3/4
B) 3/4
C) -4/3
D) 2/3

Question 58

Find $\lim _ { x \rightarrow 3 ^ { - } } \frac { 2 x } { x ^ { 2 } - 9 }$

Question 59

Find $\lim _ { x \rightarrow 4 } \frac { 4 - x } { \sqrt { x } - 2 }$

A) 4
B) -4
C) 0
D) Does not exist

Question 60

Find $\lim _ { x \rightarrow 6 } \frac { \frac { 1 } { x } - \frac { 1 } { 6 } } { x - 6 }$

A) -1/36
B) 1/36
C) -1/12
D) Does not exist

Question 61

Find $\lim _ { x \rightarrow 4 } \frac { \frac { 3 } { 4 } - \frac { 3 } { x } } { x - 4 }$

Question 62

Find $\lim _ { h \rightarrow 0 } \frac { \frac { 1 } { 3 + h } - \frac { 1 } { 3 } } { h }$

A) 1/3
B) 1/9
C) -1/9
D) Does not exist

Question 63

Find $\lim _ { x \rightarrow 0 } \frac { e ^ { 2 x } - 1 } { x }$

A) 1
B) -2
C) + $\infty$
D) 2

Question 64

Find $\lim _ { x \rightarrow ( \pi / 2 ) ^ { - } } \frac { 3 + 2 \cos x } { \tan x }$

A) - $\infty$
B) + $\infty$
C) Does not exist
D) 0

Question 65

Find $\lim _ { x \rightarrow \pi } \frac { 4 \tan x } { \sin x }$

Question 66

Find $\lim _ { x \rightarrow 0 ^ { - } } \frac { 3 - \cos x } { x }$

A) - $\infty$
B) + $\infty$
C) Does not exist
D) 1

Question 67

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 3 \cos x }$

Question 68

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 2 } { 3 \cos x }$

A) 2/3
B) 0
C) Does not exist
D) -1/3

Question 69

Find $\lim _ { x \rightarrow 0 } \frac { 3 - \cos x } { x }$

A) - $\infty$
B) + $\infty$
C) Does not exist
D) 0

Question 70

Find $\lim _ { x \rightarrow 0 } \frac { 1 - \cos 5 x } { x }$

Question 71

Find $\lim _ { x \rightarrow 0 } \frac { 5 x } { \sin 3 x }$

A) 3/5
B) 5/3
C) 1
D) Does not exist

Question 72

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 5 x }$

Question 73

Find $\lim _ { x \rightarrow 0 } \frac { \cos x - 1 } { 2 \sin x }$

A) 1/2
B) 0
C) 1
D) Does not exist

Question 74

Find $\lim _ { x \rightarrow 0 } ( 1 + x ) ^ { 2 / x }$

A) $e ^ { 3 }$
B) e
C) $e ^ { 2 }$
D) 1

Question 75

Find $\lim _ { x \rightarrow \infty } \left( 1 + \frac { 5 } { x } \right) ^ { 3 x }$

A) e^-12
B) e
C) e¹⁵
D) 1

Question 76

Find $\lim _ { x \rightarrow 1 } 2 x \cos ^ { - 1 } \left( \frac { x } { 2 } \right)$

Question 77

Find $\lim _ { x \rightarrow 1 } \frac { 5 x } { \tan ^ { - 1 } \sqrt { x } }$

Question 78

Find $\lim _ { x \rightarrow 1 } \frac { \tan ^ { - 1 } ( \sqrt { x } ) } { 3 x }$

Question 79

Find $\lim _ { x \rightarrow + \infty } \frac { \tan ^ { - 1 } ( \sqrt { x } ) } { 3 x }$

Question 80

Find $\lim _ { x \rightarrow - \infty } \frac { 1 } { 2 \tan ^ { - 1 } x }$