Question 1

Which of the following is equal to limit $\lim _ { x \rightarrow 0 ^ { - } } - \frac { 2 x } { \sin ( 3 x ) }$ ?

Accepted Answer

A)  $- \frac { 3 } { 2 }$ 
B)  $- \frac { 2 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 3 } { 2 }$ 
E) Does not exist 
A)  $- \frac { 3 } { 2 }$ 
B)  $- \frac { 2 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $\frac { 3 } { 2 }$ 
E) Does not exist

Question 2

Let $f ( x ) = \frac { \sqrt { x + 3 } - \sqrt { 3 } } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 } f ( x )$ ?

Accepted Answer

A)  $- \frac { \sqrt { 3 } } { 6 }$ 
B) 0 
C)  $\frac { \sqrt { 3 } } { 6 }$ 
D) 1 
E) Does not exist 
A)  $- \frac { \sqrt { 3 } } { 6 }$ 
B) 0 
C)  $\frac { \sqrt { 3 } } { 6 }$ 
D) 1 
E) Does not exist

Question 3

The limit $\lim _ { x \rightarrow \frac { \pi } { 2 } ^ { + } } \sec x$ is

Accepted Answer

A)  $- \infty$ 
B)  $\infty$ 
C) -1 
D) 1 
E) 0 
A)  $- \infty$ 
B)  $\infty$ 
C) -1 
D) 1 
E) 0

Question 4

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

Accepted Answer

A) ƒ is undefined at 0. 
B) ƒ is undefined at $- \frac { 1 } { 2 }$ 
C) ƒ has a removable discontinuity at 0. 
D) ƒ has a removable discontinuity at $\frac { 1 } { 2 }$ 
E) ƒ has a jump discontinuity at 0. 
A) ƒ is undefined at 0. 
B) ƒ is undefined at $- \frac { 1 } { 2 }$ 
C) ƒ has a removable discontinuity at 0. 
D) ƒ has a removable discontinuity at $\frac { 1 } { 2 }$ 
E) ƒ has a jump discontinuity at 0.

Question 5

Consider $\lim _ { x \rightarrow r } \sin ( x - \pi ) = 0$ In order for $| \sin ( x - \pi ) - 0 | < 0.01$ whenever $0 < | x - \pi | < \delta$ , which of the following is the largest ? of those listed for which this is true?

Accepted Answer

A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) 0.10 
A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) 0.10

Question 6

Consider $\lim _ { x \rightarrow - 5 } \left( \frac { x } { 5 } - 3 \right) = - 4$ In order for $\left| \left( \frac { x } { 5 } - 3 \right) - ( - 4 ) \right| < 0.01$ whenever $0 < | x - ( - 5 ) | < \delta$ which of the following is true for the largest ?

Accepted Answer

A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) Impossible to find 
A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) Impossible to find

Question 7

The limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { x ^ { 2 } - 2 } { x ^ { 3 } + x ^ { 2 } }$ is

Accepted Answer

A)  $- \infty$ 
B)  $\infty$ 
C) -2 
D) 2 
E) -4 
A)  $- \infty$ 
B)  $\infty$ 
C) -2 
D) 2 
E) -4

Question 8

Consider $\lim _ { x \rightarrow 0 } \frac { 1 } { e } = \frac { 1 } { e }$ In order for $\left| \frac { 1 } { e } - \frac { 1 } { e } \right| < 0.01$ whenever $0 < | x - 0 | < \delta$ which of the following is true for the largest ?

Accepted Answer

A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) Impossible to find 
A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) Impossible to find

Question 9

Consider $\lim _ { x \rightarrow - 7 } \left( \frac { 2 x } { 7 } - 5 \right) = - 7$ In order for $\left| \left( \frac { 2 x } { 7 } - 5 \right) - ( - 7 ) \right| < 0.01$ whenever $0 < | x - ( - 7 ) | < \delta$ which of the following is true for the largest ?

Accepted Answer

A) 0.001 
B) 0.025 
C) 0.03 
D) 0.035 
E) Impossible to find 
A) 0.001 
B) 0.025 
C) 0.03 
D) 0.035 
E) Impossible to find

Question 10

Let $f ( x ) = \left\{ \begin{array} { c c } 
- x ^ { 2 } + 2 & - 3 < x < 0 \
3 & x = 0 \
2 x + 4 & 0 < x \leq 3
\end{array} \right.$ Which of the following is true for $\lim _ { x \rightarrow 0 } f ( x )$ ?

Accepted Answer

A) 2 
B) 3 
C) 4 
D) 0 
E) Does not exist 
A) 2 
B) 3 
C) 4 
D) 0 
E) Does not exist

Question 11

Consider $\lim _ { x \rightarrow 2 } 6 = 6$ In order for $| 6 - 6 | < 0.01$ whenever $0 < | x - 2 | < \delta$ , which of the following is true for the largest ?

Accepted Answer

A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) Impossible to find 
A) 0.001 
B) 0.005 
C) 0.01 
D) 0.05 
E) Impossible to find

Question 12

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 ^ { - } } f ( x )$ ?

Accepted Answer

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

Question 13

Let $f ( x ) = \sqrt { x + 1 }$ Which of the following is true for $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ ?

Accepted Answer

A) -1 
B) 0 
C) 1 
D)  $\sqrt { 2 }$ 
E) Does not exist 
A) -1 
B) 0 
C) 1 
D)  $\sqrt { 2 }$ 
E) Does not exist

Question 14

Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \frac { - 3 x ^ { 3 } + 4 \sin ( 6 x ) } { 5 x }$ ?

Accepted Answer

A)  $- \frac { 4 } { 5 }$ 
B)  $- \frac { 4 } { 5 }$. 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 24 } { 5 }$ 
E) Does not exist 
A)  $- \frac { 4 } { 5 }$ 
B)  $- \frac { 4 } { 5 }$. 
C)  $\frac { 4 } { 5 }$ 
D)  $\frac { 24 } { 5 }$ 
E) Does not exist

Question 15

Let $f ( x ) = \left\{ \begin{array} { c c } 
x ^ { 2 } & - 3 \leq x < 0 \
3 x & 0 < x < 3 \
2 x - 5 & 3 < x < 10
\end{array} \right.$ Then ƒ is continuous at 0 if ƒ(0) is defined as

Accepted Answer

A) -2 
B) -1 
C) 0 
D) 1 
E) 2 
A) -2 
B) -1 
C) 0 
D) 1 
E) 2

Question 16

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ ?

Accepted Answer

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

Question 17

The limit $\lim _ { x \rightarrow - \infty } \frac { 4 - x } { x ^ { 2 } - 5 }$ is

Accepted Answer

A)  $- \infty$ 
B)  $\infty$ 
C)  $- \frac { 4 } { 5 }$ 
D) 0 
E)  $\frac { 4 } { 5 }$ 
A)  $- \infty$ 
B)  $\infty$ 
C)  $- \frac { 4 } { 5 }$ 
D) 0 
E)  $\frac { 4 } { 5 }$

Question 18

Let $f ( x ) = \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }$ . Which of the following is true?

Accepted Answer

A) ƒ is defined at -2. 
B) ƒ is defined at 2. 
C) ƒ has a removable discontinuity at -2. 
D) ƒ has a removable discontinuity at 2. 
E) ƒ is continuous at -2. 
A) ƒ is defined at -2. 
B) ƒ is defined at 2. 
C) ƒ has a removable discontinuity at -2. 
D) ƒ has a removable discontinuity at 2. 
E) ƒ is continuous at -2.

Question 19

Let $f ( x ) = \sec x$ for $x \in [ 0,2 \pi ]$ The set of all vertical asymptotes of ƒ is

Accepted Answer

A)  $\left\{ x = \frac { \pi } { 2 } \right\}$ 
B)  $\left\{ x = \frac { 3 \pi } { 2 } \right\}$ 
C)  $\left\{ x = - \frac { \pi } { 2 } \right\}$ 
D)  $\left\{ x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 } \right\}$ 
E)  $\left\{ x = - \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 } \right\}$ 
A)  $\left\{ x = \frac { \pi } { 2 } \right\}$ 
B)  $\left\{ x = \frac { 3 \pi } { 2 } \right\}$ 
C)  $\left\{ x = - \frac { \pi } { 2 } \right\}$ 
D)  $\left\{ x = \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 } \right\}$ 
E)  $\left\{ x = - \frac { \pi } { 2 } , x = \frac { 3 \pi } { 2 } \right\}$

Question 20

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = $\frac { x ^ { 3 } + 1 } { x - 2 }$ and y = -1.9 in the interval (-3, 3).

Accepted Answer

A) -2.025 
B) 0.979 
C) 1.005 
D) 1.502 
E) 2.025 
A) -2.025 
B) 0.979 
C) 1.005 
D) 1.502 
E) 2.025

Which of the following is equal to limit $\lim _ { x \rightarrow 0 ^ { - } } - \frac { 2 x } { \sin ( 3 x ) }$ ?

Let $f ( x ) = \frac { \sqrt { x + 3 } - \sqrt { 3 } } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 } f ( x )$ ?

The limit $\lim _ { x \rightarrow \frac { \pi } { 2 } ^ { + } } \sec x$ is

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

Consider $\lim _ { x \rightarrow r } \sin ( x - \pi ) = 0$ In order for $| \sin ( x - \pi ) - 0 | < 0.01$ whenever $0 < | x - \pi | < \delta$ , which of the following is the largest ? of those listed for which this is true?

Consider $\lim _ { x \rightarrow - 5 } \left( \frac { x } { 5 } - 3 \right) = - 4$ In order for $\left| \left( \frac { x } { 5 } - 3 \right) - ( - 4 ) \right| < 0.01$ whenever $0 < | x - ( - 5 ) | < \delta$ which of the following is true for the largest ?

The limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { x ^ { 2 } - 2 } { x ^ { 3 } + x ^ { 2 } }$ is

Consider $\lim _ { x \rightarrow 0 } \frac { 1 } { e } = \frac { 1 } { e }$ In order for $\left| \frac { 1 } { e } - \frac { 1 } { e } \right| < 0.01$ whenever $0 < | x - 0 | < \delta$ which of the following is true for the largest ?

Consider $\lim _ { x \rightarrow - 7 } \left( \frac { 2 x } { 7 } - 5 \right) = - 7$ In order for $\left| \left( \frac { 2 x } { 7 } - 5 \right) - ( - 7 ) \right| < 0.01$ whenever $0 < | x - ( - 7 ) | < \delta$ which of the following is true for the largest ?

Let $f ( x ) = \left\{ \begin{array} { c c } - x ^ { 2 } + 2 & - 3 < x < 0 \\3 & x = 0 \\2 x + 4 & 0 < x \leq 3\end{array} \right.$ Which of the following is true for $\lim _ { x \rightarrow 0 } f ( x )$ ?

Consider $\lim _ { x \rightarrow 2 } 6 = 6$ In order for $| 6 - 6 | < 0.01$ whenever $0 < | x - 2 | < \delta$ , which of the following is true for the largest ?

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 ^ { - } } f ( x )$ ?

Let $f ( x ) = \sqrt { x + 1 }$ Which of the following is true for $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ ?

Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \frac { - 3 x ^ { 3 } + 4 \sin ( 6 x ) } { 5 x }$ ?

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & - 3 \leq x < 0 \\3 x & 0 < x < 3 \\2 x - 5 & 3 < x < 10\end{array} \right.$ Then ƒ is continuous at 0 if ƒ(0) is defined as

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ ?

The limit $\lim _ { x \rightarrow - \infty } \frac { 4 - x } { x ^ { 2 } - 5 }$ is

Let $f ( x ) = \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }$ . Which of the following is true?

Let $f ( x ) = \sec x$ for $x \in [ 0,2 \pi ]$ The set of all vertical asymptotes of ƒ is

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = $\frac { x ^ { 3 } + 1 } { x - 2 }$ and y = -1.9 in the interval (-3, 3).

Preparing for Calculus

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

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Exam 2: Limits and Continuity

Which of the following is equal to limit lim⁡x→0−−2xsin⁡(3x)\lim _ { x \rightarrow 0 ^ { - } } - \frac { 2 x } { \sin ( 3 x ) }limx→0−​−sin(3x)2x​ ?

Let f(x)=x+3−3xf ( x ) = \frac { \sqrt { x + 3 } - \sqrt { 3 } } { x }f(x)=xx+3​−3​​ Which of the following is true for lim⁡x→0f(x)\lim _ { x \rightarrow 0 } f ( x )limx→0​f(x) ?

The limit lim⁡x→π2+sec⁡x\lim _ { x \rightarrow \frac { \pi } { 2 } ^ { + } } \sec xlimx→2π​+​secx is

Let f(x)=⌊x⌋f ( x ) = \lfloor x \rfloorf(x)=⌊x⌋ Which of the following is true?

The limit lim⁡x→0+x2−2x3+x2\lim _ { x \rightarrow 0 ^ { + } } \frac { x ^ { 2 } - 2 } { x ^ { 3 } + x ^ { 2 } }limx→0+​x3+x2x2−2​ is

Consider lim⁡x→26=6\lim _ { x \rightarrow 2 } 6 = 6limx→2​6=6 In order for ∣6−6∣<0.01| 6 - 6 | < 0.01∣6−6∣<0.01 whenever 0<∣x−2∣<δ0 < | x - 2 | < \delta0<∣x−2∣<δ , which of the following is true for the largest ?

Let f(x)=∣x∣xf ( x ) = \frac { | x | } { x }f(x)=x∣x∣​ Which of the following is true for lim⁡x→0−f(x)\lim _ { x \rightarrow 0 ^ { - } } f ( x )limx→0−​f(x) ?

Let f(x)=x+1f ( x ) = \sqrt { x + 1 }f(x)=x+1​ Which of the following is true for lim⁡x→1+f(x)\lim _ { x \rightarrow 1 ^ { + } } f ( x )limx→1+​f(x) ?

Which of the following is true for the limit lim⁡x→0−3x3+4sin⁡(6x)5x\lim _ { x \rightarrow 0 } \frac { - 3 x ^ { 3 } + 4 \sin ( 6 x ) } { 5 x }limx→0​5x−3x3+4sin(6x)​ ?

Let f(x)={x2−3≤x<03x0<x<32x−53<x<10f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & - 3 \leq x < 0 \\3 x & 0 < x < 3 \\2 x - 5 & 3 < x < 10\end{array} \right.f(x)=⎩⎨⎧​x23x2x−5​−3≤x<00<x<33<x<10​ Then ƒ is continuous at 0 if ƒ(0) is defined as

Let f(x)=∣x∣xf ( x ) = \frac { | x | } { x }f(x)=x∣x∣​ Which of the following is true for lim⁡x→0+f(x)\lim _ { x \rightarrow 0 ^ { + } } f ( x )limx→0+​f(x) ?

The limit lim⁡x→−∞4−xx2−5\lim _ { x \rightarrow - \infty } \frac { 4 - x } { x ^ { 2 } - 5 }limx→−∞​x2−54−x​ is

Let f(x)=x3−8x2−4f ( x ) = \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }f(x)=x2−4x3−8​ . Which of the following is true?

Let f(x)=sec⁡xf ( x ) = \sec xf(x)=secx for x∈[0,2π]x \in [ 0,2 \pi ]x∈[0,2π] The set of all vertical asymptotes of ƒ is

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = x3+1x−2\frac { x ^ { 3 } + 1 } { x - 2 }x−2x3+1​ and y = -1.9 in the interval (-3, 3).

Preparing for Calculus

The Derivative

More About Derivatives

Applications of the Derivative

The Integral

Applications of the Integral

Techniques of Integration

Infinite Series

Parametric Equations; Polar Equations

Vectors; Lines, Planes, and Quadric Surfaces in Space

Vector Functions

Functions of Several Variables

Directional Derivatives, Gradients, and Extrema

Multiple Integrals

Vector Calculus

Differential Equations

Filters

Which of the following is equal to limit $\lim _ { x \rightarrow 0 ^ { - } } - \frac { 2 x } { \sin ( 3 x ) }$ ?

Let $f ( x ) = \frac { \sqrt { x + 3 } - \sqrt { 3 } } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 } f ( x )$ ?

The limit $\lim _ { x \rightarrow \frac { \pi } { 2 } ^ { + } } \sec x$ is

Let $f ( x ) = \lfloor x \rfloor$ Which of the following is true?

The limit $\lim _ { x \rightarrow 0 ^ { + } } \frac { x ^ { 2 } - 2 } { x ^ { 3 } + x ^ { 2 } }$ is

Consider $\lim _ { x \rightarrow 2 } 6 = 6$ In order for $| 6 - 6 | < 0.01$ whenever $0 < | x - 2 | < \delta$ , which of the following is true for the largest ?

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 ^ { - } } f ( x )$ ?

Let $f ( x ) = \sqrt { x + 1 }$ Which of the following is true for $\lim _ { x \rightarrow 1 ^ { + } } f ( x )$ ?

Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \frac { - 3 x ^ { 3 } + 4 \sin ( 6 x ) } { 5 x }$ ?

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & - 3 \leq x < 0 \\3 x & 0 < x < 3 \\2 x - 5 & 3 < x < 10\end{array} \right.$ Then ƒ is continuous at 0 if ƒ(0) is defined as

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true for $\lim _ { x \rightarrow 0 ^ { + } } f ( x )$ ?

The limit $\lim _ { x \rightarrow - \infty } \frac { 4 - x } { x ^ { 2 } - 5 }$ is

Let $f ( x ) = \frac { x ^ { 3 } - 8 } { x ^ { 2 } - 4 }$ . Which of the following is true?

Let $f ( x ) = \sec x$ for $x \in [ 0,2 \pi ]$ The set of all vertical asymptotes of ƒ is

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = $\frac { x ^ { 3 } + 1 } { x - 2 }$ and y = -1.9 in the interval (-3, 3).