Question 1

Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \frac { e ^ { 2 x } - 1 } { 1 - e ^ { 3 x } }$ ?

Accepted Answer

A)  $- \frac { 2 } { 3 }$ 
B)  $- \frac { 3 } { 2 }$ 
C) 0 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 3 } { 2 }$ 
A)  $- \frac { 2 } { 3 }$ 
B)  $- \frac { 3 } { 2 }$ 
C) 0 
D)  $\frac { 2 } { 3 }$ 
E)  $\frac { 3 } { 2 }$ A

Question 2

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

Accepted Answer

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

Question 3

Let $f ( x ) = \left\{ \begin{array} { c c } 
2 x - 5 & x  4
\end{array} \right.$ Then is continuous at 1 if ƒ(4) is redefined as

Accepted Answer

A) -1 
B) 0 
C) 1 
D) 2 
E) 3 
A) -1 
B) 0 
C) 1 
D) 2 
E) 3 E

Question 4

Let $f ( x ) = 2 \lfloor x \rfloor - 3$ Which of the following is true for $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ ?

Accepted Answer

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

Question 5

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 - e ^ { x } } { 1 + e ^ { 3 x } } \right)$ ?

Accepted Answer

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

Question 6

Consider $\lim _ { x \rightarrow \frac { \pi } { 2 } } \sin \left( x - \frac { \pi } { 2 } \right) = 0$ In order for $\left| \sin \left( x - \frac { \pi } { 2 } \right) - 0 \right| < 0.01$ whenever $0 < \left| x - \frac { \pi } { 2 } \right| < \delta$ , 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

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 \sin ( 2 x ) } { 5 x }$ ?

Accepted Answer

A)  $- \frac { 6 } { 5 }$ 
B)  $- \frac { 5 } { 6 }$ 
C) 0 
D)  $\frac { 5 } { 6 }$ 
E)  $\frac { 1 } { 5 }$ 
A)  $- \frac { 6 } { 5 }$ 
B)  $- \frac { 5 } { 6 }$ 
C) 0 
D)  $\frac { 5 } { 6 }$ 
E)  $\frac { 1 } { 5 }$

Question 8

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \left( \frac { 4 - e ^ { 2 x } } { 5 + e ^ { 3 x } } \right)$ ?

Accepted Answer

A)  $\frac { 4 } { 5 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E) Does not exist 
A)  $\frac { 4 } { 5 }$ 
B)  $\frac { 2 } { 3 }$ 
C)  $\frac { 1 } { 2 }$ 
D) 0 
E) Does not exist

Question 9

Consider $\lim _ { x \rightarrow - x } \cos ( x - \pi ) = 1$ In order for $| \cos ( x - \pi ) - 1 | < 0.01$ 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 10

Consider $\lim _ { x \rightarrow 2 } ( 6 x - 1 ) = 11$ In order for $| ( 6 x - 1 ) - 11 | < 0.03$ whenever $0 < | x - 2 | < \delta$ which of the following is true for the largest ??

Accepted Answer

A) 0.001 
B) 0.005 
C) 0.03 
D) 0.05 
E) Impossible to find 
A) 0.001 
B) 0.005 
C) 0.03 
D) 0.05 
E) Impossible to find

Question 11

Let $f ( x ) = \left\{ \begin{array} { c c } 
\frac { 9 } { x } & - 5 < x < - 3 \
2 & x = - 3 \
x & - 3 < x < 1
\end{array} \right.$ Which of the following is true for $\lim _ { x \rightarrow - 3 } f ( x )$ ?

Accepted Answer

A) -3 
B) 2 
C) 3 
D) 5 
E) Does not exist 
A) -3 
B) 2 
C) 3 
D) 5 
E) Does not exist

Question 12

Which of the following equal to limit $\lim _ { x \rightarrow 1 } \ln \left( \frac { x ^ { 4 } } { e ^ { 4 x } } \right)$ ?

Accepted Answer

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

Question 13

Consider $\lim _ {x \rightarrow - 3 } ( 2 | x | + 3 ) = 9$ In order for $| ( 2 | x | + 3 ) - 9 | < 0.01$ whenever $0 < \mid x - ( - 3 x \mid <\delta =$  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 14

Let $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }$ Which of the following is true?

Accepted Answer

A) ƒ is defined at 1. 
B) ƒ is defined at -1. 
C) ƒ has a removable discontinuity at -1. 
D) ƒ has a removable discontinuity at 1. 
E) ƒ is continuous at -1. 
A) ƒ is defined at 1. 
B) ƒ is defined at -1. 
C) ƒ has a removable discontinuity at -1. 
D) ƒ has a removable discontinuity at 1. 
E) ƒ is continuous at -1.

Question 15

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

Accepted Answer

A) 2 
B) 3 
C) 4 
D) 5 
E) Does not exist 
A) 2 
B) 3 
C) 4 
D) 5 
E) Does not exist

Question 16

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true?

Accepted Answer

A) ƒ is defined at 0. 
B) ƒ is discontinuous at -1. 
C) ƒ has a removable discontinuity at 0. 
D) ƒ has a removable discontinuity at -1. 
E) ƒ has a jump discontinuity at 0. 
A) ƒ is defined at 0. 
B) ƒ is discontinuous at -1. 
C) ƒ has a removable discontinuity at 0. 
D) ƒ has a removable discontinuity at -1. 
E) ƒ has a jump discontinuity at 0.

Question 17

Let $f ( x ) = \frac { \sqrt { x } + 4 } { 2 x - 3 }$ The set of all horizontal asymptotes of ƒ is

Accepted Answer

A)  $\left\{ y = - \frac { 1 } { 2 } \right\}$ 
B)  $\{ y = 0 \}$ 
C)  $\left\{ y = \frac { 1 } { 2 } \right\}$ 
D)  $\left\{ y = - \frac { 1 } { 2 } , y = \frac { 1 } { 2 } \right\}$ 
E)  $\varnothing$ 
A)  $\left\{ y = - \frac { 1 } { 2 } \right\}$ 
B)  $\{ y = 0 \}$ 
C)  $\left\{ y = \frac { 1 } { 2 } \right\}$ 
D)  $\left\{ y = - \frac { 1 } { 2 } , y = \frac { 1 } { 2 } \right\}$ 
E)  $\varnothing$

Question 18

Let $f ( x ) = \frac { x ^ { 3 } - 27 } { x ^ { 2 } - 9 }$ Which of the following is true for $\lim _ { x \rightarrow 3 } f ( x )$ ?

Accepted Answer

A) 9 
B)  $\frac { 9 } { 2 }$ 
C) -9 
D)  $- \frac { 9 } { 2 }$ 
E) Does not exist 
A) 9 
B)  $\frac { 9 } { 2 }$ 
C) -9 
D)  $- \frac { 9 } { 2 }$ 
E) Does not exist

Question 19

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \frac { \cos x } { e ^ { x } }$ ?

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 20

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \ln \left( \frac { e ^ { x } } { 2 } \right)$ ?

Accepted Answer

A)  $\ln 2$ 
B) 0 
C)  $\ln 2$. 
D)  $\frac { 1 } { 2 }$ 
E) Does not exist 
A)  $\ln 2$ 
B) 0 
C)  $\ln 2$. 
D)  $\frac { 1 } { 2 }$ 
E) Does not exist

Which of the following is true for the limit $\lim _ { x \rightarrow 0 } \frac { e ^ { 2 x } - 1 } { 1 - e ^ { 3 x } }$ ?

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

Let $f ( x ) = \left\{ \begin{array} { c c } 2 x - 5 & x < 4 \\4 & x = 4 \\x - 1 & x > 4\end{array} \right.$ Then is continuous at 1 if ƒ(4) is redefined as

Let $f ( x ) = 2 \lfloor x \rfloor - 3$ Which of the following is true for $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 - e ^ { x } } { 1 + e ^ { 3 x } } \right)$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 \sin ( 2 x ) } { 5 x }$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \left( \frac { 4 - e ^ { 2 x } } { 5 + e ^ { 3 x } } \right)$ ?

Consider $\lim _ { x \rightarrow - x } \cos ( x - \pi ) = 1$ In order for $| \cos ( x - \pi ) - 1 | < 0.01$ whenever $0 < | x - ( - \pi ) | < \delta$ which of the following is the largest ? of those listed for which this is true?

Consider $\lim _ { x \rightarrow 2 } ( 6 x - 1 ) = 11$ In order for $| ( 6 x - 1 ) - 11 | < 0.03$ whenever $0 < | x - 2 | < \delta$ which of the following is true for the largest ??

Let $f ( x ) = \left\{ \begin{array} { c c } \frac { 9 } { x } & - 5 < x < - 3 \\2 & x = - 3 \\x & - 3 < x < 1\end{array} \right.$ Which of the following is true for $\lim _ { x \rightarrow - 3 } f ( x )$ ?

Which of the following equal to limit $\lim _ { x \rightarrow 1 } \ln \left( \frac { x ^ { 4 } } { e ^ { 4 x } } \right)$ ?

Consider $\lim _ {x \rightarrow - 3 } ( 2 | x | + 3 ) = 9$ In order for $| ( 2 | x | + 3 ) - 9 | < 0.01$ whenever $0 < \mid x - ( - 3 x \mid <\delta =$ which of the following is true for the largest ?

Let $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }$ Which of the following is true?

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

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true?

Let $f ( x ) = \frac { \sqrt { x } + 4 } { 2 x - 3 }$ The set of all horizontal asymptotes of ƒ is

Let $f ( x ) = \frac { x ^ { 3 } - 27 } { x ^ { 2 } - 9 }$ Which of the following is true for $\lim _ { x \rightarrow 3 } f ( x )$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \frac { \cos x } { e ^ { x } }$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \ln \left( \frac { e ^ { x } } { 2 } \right)$ ?

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

Exam 2: Limits and Continuity

Which of the following is true for the limit lim⁡x→0e2x−11−e3x\lim _ { x \rightarrow 0 } \frac { e ^ { 2 x } - 1 } { 1 - e ^ { 3 x } }limx→0​1−e3xe2x−1​ ?

Let f(x)=1x−11−xf ( x ) = \frac { \frac { 1 } { x } - 1 } { 1 - x }f(x)=1−xx1​−1​ Which of the following is true for lim⁡x→1f(x)\lim _ { x \rightarrow 1 } f ( x )limx→1​f(x) ?

Let f(x)={2x−5x<44x=4x−1x>4f ( x ) = \left\{ \begin{array} { c c } 2 x - 5 & x < 4 \\4 & x = 4 \\x - 1 & x > 4\end{array} \right.f(x)=⎩⎨⎧​2x−54x−1​x<4x=4x>4​ Then is continuous at 1 if ƒ(4) is redefined as

Let f(x)=2⌊x⌋−3f ( x ) = 2 \lfloor x \rfloor - 3f(x)=2⌊x⌋−3 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 equal to limit lim⁡x→0(1−ex1+e3x)\lim _ { x \rightarrow 0 } \left( \frac { 1 - e ^ { x } } { 1 + e ^ { 3 x } } \right)limx→0​(1+e3x1−ex​) ?

Which of the following is equal to limit lim⁡x→0x2−3sin⁡(2x)5x\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 \sin ( 2 x ) } { 5 x }limx→0​5xx2−3sin(2x)​ ?

Which of the following is equal to limit lim⁡x→0(4−e2x5+e3x)\lim _ { x \rightarrow 0 } \left( \frac { 4 - e ^ { 2 x } } { 5 + e ^ { 3 x } } \right)limx→0​(5+e3x4−e2x​) ?

Consider lim⁡x→2(6x−1)=11\lim _ { x \rightarrow 2 } ( 6 x - 1 ) = 11limx→2​(6x−1)=11 In order for ∣(6x−1)−11∣<0.03| ( 6 x - 1 ) - 11 | < 0.03∣(6x−1)−11∣<0.03 whenever 0<∣x−2∣<δ0 < | x - 2 | < \delta0<∣x−2∣<δ which of the following is true for the largest ??

Which of the following equal to limit lim⁡x→1ln⁡(x4e4x)\lim _ { x \rightarrow 1 } \ln \left( \frac { x ^ { 4 } } { e ^ { 4 x } } \right)limx→1​ln(e4xx4​) ?

Let f(x)=x+1x2−1f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }f(x)=x2−1x+1​ Which of the following is true?

Let f(x)=⌊x⌋+3f ( x ) = \lfloor x \rfloor + 3f(x)=⌊x⌋+3 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∣xf ( x ) = \frac { | x | } { x }f(x)=x∣x∣​ Which of the following is true?

Let f(x)=x+42x−3f ( x ) = \frac { \sqrt { x } + 4 } { 2 x - 3 }f(x)=2x−3x​+4​ The set of all horizontal asymptotes of ƒ is

Let f(x)=x3−27x2−9f ( x ) = \frac { x ^ { 3 } - 27 } { x ^ { 2 } - 9 }f(x)=x2−9x3−27​ Which of the following is true for lim⁡x→3f(x)\lim _ { x \rightarrow 3 } f ( x )limx→3​f(x) ?

Which of the following is equal to limit lim⁡x→0cos⁡xex\lim _ { x \rightarrow 0 } \frac { \cos x } { e ^ { x } }limx→0​excosx​ ?

Which of the following is equal to limit lim⁡x→0ln⁡(ex2)\lim _ { x \rightarrow 0 } \ln \left( \frac { e ^ { x } } { 2 } \right)limx→0​ln(2ex​) ?

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 true for the limit $\lim _ { x \rightarrow 0 } \frac { e ^ { 2 x } - 1 } { 1 - e ^ { 3 x } }$ ?

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

Let $f ( x ) = \left\{ \begin{array} { c c } 2 x - 5 & x < 4 \\4 & x = 4 \\x - 1 & x > 4\end{array} \right.$ Then is continuous at 1 if ƒ(4) is redefined as

Let $f ( x ) = 2 \lfloor x \rfloor - 3$ Which of the following is true for $\lim _ { x \rightarrow 1 ^ { - } } f ( x )$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \left( \frac { 1 - e ^ { x } } { 1 + e ^ { 3 x } } \right)$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \frac { x ^ { 2 } - 3 \sin ( 2 x ) } { 5 x }$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \left( \frac { 4 - e ^ { 2 x } } { 5 + e ^ { 3 x } } \right)$ ?

Consider $\lim _ { x \rightarrow 2 } ( 6 x - 1 ) = 11$ In order for $| ( 6 x - 1 ) - 11 | < 0.03$ whenever $0 < | x - 2 | < \delta$ which of the following is true for the largest ??

Which of the following equal to limit $\lim _ { x \rightarrow 1 } \ln \left( \frac { x ^ { 4 } } { e ^ { 4 x } } \right)$ ?

Let $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }$ Which of the following is true?

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

Let $f ( x ) = \frac { | x | } { x }$ Which of the following is true?

Let $f ( x ) = \frac { \sqrt { x } + 4 } { 2 x - 3 }$ The set of all horizontal asymptotes of ƒ is

Let $f ( x ) = \frac { x ^ { 3 } - 27 } { x ^ { 2 } - 9 }$ Which of the following is true for $\lim _ { x \rightarrow 3 } f ( x )$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \frac { \cos x } { e ^ { x } }$ ?

Which of the following is equal to limit $\lim _ { x \rightarrow 0 } \ln \left( \frac { e ^ { x } } { 2 } \right)$ ?