Question 1

The limit $\lim _ { x \rightarrow 1 ^ { + } } \frac { \lfloor x \rfloor + 1 } { x + 1 }$ is

Accepted Answer

A)  $- \infty$ 
B) 0 
C) -2 
D) 2 
E)  $\infty$ 
A)  $- \infty$ 
B) 0 
C) -2 
D) 2 
E)  $\infty$

Question 2

The limit $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }$ is

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

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

Question 5

Let $f ( x ) = 3 x ^ { 2 } - 5$ Which of the following is equal to $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$ ?

Accepted Answer

A) -6 
B) -1 
C) 1 
D) 6 
E) Does not exist 
A) -6 
B) -1 
C) 1 
D) 6 
E) Does not exist

Question 6

Consider $\lim _ { x \rightarrow 3 } \left( \frac { x } { 4 } - 1 \right) = 1$ In order for $\left| \left( \frac { x } { 4 } - 1 \right) - 1 \right| < 0.01$ whenever $0 < | x - 8 | < \delta$ which of the following is true for the largest ?

Accepted Answer

A) 0.001 
B) 0.004 
C) 0.01 
D) 0.04 
E) Impossible to find 
A) 0.001 
B) 0.004 
C) 0.01 
D) 0.04 
E) Impossible to find

Question 7

Let $f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 2 x ^ { 2 } - x - 3 }$ Which of the following is true for $\lim _ { x \rightarrow - 1 } f ( x )$ ?

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 8

The largest set on which $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }$ is continuous is

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - \infty , 1 ) \cup ( 1 , \infty )$ 
C)  $( - \infty , - 1 ) \cup ( - 1 , \infty )$ 
D)  $( - \infty , - 1 ) \cup ( 1 , \infty )$ 
E)  $( - \infty , - 1 ) \cup ( - 1,1 ) \cup ( 1 , \infty )$ 
A)  $( - \infty , \infty )$ 
B)  $( - \infty , 1 ) \cup ( 1 , \infty )$ 
C)  $( - \infty , - 1 ) \cup ( - 1 , \infty )$ 
D)  $( - \infty , - 1 ) \cup ( 1 , \infty )$ 
E)  $( - \infty , - 1 ) \cup ( - 1,1 ) \cup ( 1 , \infty )$

Question 9

The limit is $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } + 2 x - 3 } { x - 1 }$ ?

Accepted Answer

A)  $- \infty$ 
B)  $\infty$ 
C) 2 
D) 3 
E) 4 
A)  $- \infty$ 
B)  $\infty$ 
C) 2 
D) 3 
E) 4

Question 10

The largest set on which $f ( x ) = \frac { 5 x + 3 } { 5 x ^ { 2 } + 3 }$ is continuous is

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $\left( - \infty , - \frac { 3 } { 5 } \right) \cup \left( - \frac { 3 } { 5 } , \infty \right)$ 
C)  $\left( - \infty , \frac { 3 } { 5 } \right) \cup \left( \frac { 3 } { 5 } , \infty \right)$ 
D)  $\left( - \infty , - \frac { 3 } { 5 } \right) \cup \left( \frac { 3 } { 5 } , \infty \right)$ 
E)  $\left( - \infty , - \frac { 3 } { 5 } \right) \cup \left( - \frac { 3 } { 5 } , \frac { 3 } { 5 } \right) \cup \left( \frac { 3 } { 5 } , \infty \right)$ 
A)  $( - \infty , \infty )$ 
B)  $\left( - \infty , - \frac { 3 } { 5 } \right) \cup \left( - \frac { 3 } { 5 } , \infty \right)$ 
C)  $\left( - \infty , \frac { 3 } { 5 } \right) \cup \left( \frac { 3 } { 5 } , \infty \right)$ 
D)  $\left( - \infty , - \frac { 3 } { 5 } \right) \cup \left( \frac { 3 } { 5 } , \infty \right)$ 
E)  $\left( - \infty , - \frac { 3 } { 5 } \right) \cup \left( - \frac { 3 } { 5 } , \frac { 3 } { 5 } \right) \cup \left( \frac { 3 } { 5 } , \infty \right)$

Question 11

Consider $\lim _ { x \rightarrow 4 } \left( \frac { | x | } { 2 } - 3 \right) = - 1$ In order for $\left| \left( \frac { | x | } { 2 } - 3 \right) - ( - 1 ) \right| < 0.01$ whenever $0 < | x - 4 | <\delta $ , which of the following is true for the largest ?

Accepted Answer

A) 0.001 
B) 0.002 
C) 0.01 
D) 0.02 
E) Impossible to find 
A) 0.001 
B) 0.002 
C) 0.01 
D) 0.02 
E) Impossible to find

Question 12

The limit $\lim _ { x \rightarrow 0 ^ { - } } \ln \left( e ^ { x } - 1 \right)$ is

Accepted Answer

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

Question 13

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

Accepted Answer

A)  . $f$ is defined at -1 
B)  $f$ is defined at 0. 
C)  $f$ has a removable discontinuity at -1. 
D)  $f$ has a removable discontinuity at 0. 
E)  $f$ is continuous at 0. 
A)  . $f$ is defined at -1 
B)  $f$ is defined at 0. 
C)  $f$ has a removable discontinuity at -1. 
D)  $f$ has a removable discontinuity at 0. 
E)  $f$ is continuous at 0.

Question 14

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 15

Let $f ( x ) = \frac { \frac { 3 } { x + 4 } - 1 } { 1 + x }$ Which of the following is true for $\lim _ { x \rightarrow - 1 } f ( x )$ ?

Accepted Answer

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

Question 16

The limit $\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor - 2 } { x - 2 }$ is

Accepted Answer

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

Question 17

The limit $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }$ is

Accepted Answer

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

Question 18

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

Accepted Answer

A) -5 
B) -4 
C) -3 
D) -2 
E) -1 
A) -5 
B) -4 
C) -3 
D) -2 
E) -1

Question 19

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = x4- 2x and y = 1 in the interval (-1, 0).
​

Accepted Answer

A) -1.0 
B) -0.602 
C) -0.475 
D) 0.515 
E) 1.392 
A) -1.0 
B) -0.602 
C) -0.475 
D) 0.515 
E) 1.392

Question 20

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

Accepted Answer

A) 0.001 
B) 0.005 
C) 0.003 
D) 0.05 
E) Impossible to find 
A) 0.001 
B) 0.005 
C) 0.003 
D) 0.05 
E) Impossible to find

The limit $\lim _ { x \rightarrow 1 ^ { + } } \frac { \lfloor x \rfloor + 1 } { x + 1 }$ is

The limit $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }$ is

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

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

Let $f ( x ) = 3 x ^ { 2 } - 5$ Which of the following is equal to $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$ ?

Consider $\lim _ { x \rightarrow 3 } \left( \frac { x } { 4 } - 1 \right) = 1$ In order for $\left| \left( \frac { x } { 4 } - 1 \right) - 1 \right| < 0.01$ whenever $0 < | x - 8 | < \delta$ which of the following is true for the largest ?

Let $f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 2 x ^ { 2 } - x - 3 }$ Which of the following is true for $\lim _ { x \rightarrow - 1 } f ( x )$ ?

The largest set on which $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }$ is continuous is

The limit is $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } + 2 x - 3 } { x - 1 }$ ?

The largest set on which $f ( x ) = \frac { 5 x + 3 } { 5 x ^ { 2 } + 3 }$ is continuous is

Consider $\lim _ { x \rightarrow 4 } \left( \frac { | x | } { 2 } - 3 \right) = - 1$ In order for $\left| \left( \frac { | x | } { 2 } - 3 \right) - ( - 1 ) \right| < 0.01$ whenever $0 < | x - 4 | <\delta$ , which of the following is true for the largest ?

The limit $\lim _ { x \rightarrow 0 ^ { - } } \ln \left( e ^ { x } - 1 \right)$ is

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

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

Let $f ( x ) = \frac { \frac { 3 } { x + 4 } - 1 } { 1 + x }$ Which of the following is true for $\lim _ { x \rightarrow - 1 } f ( x )$ ?

The limit $\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor - 2 } { x - 2 }$ is

The limit $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }$ is

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 5 x & x < - 1 \\0 & x = - 1 \\x - 3 & x > - 1\end{array} \right.$ Then ƒ is continuous at -1 if ƒ(-1) is redefined as

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = x⁴- 2x and y = 1 in the interval (-1, 0).

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

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

The limit lim⁡x→1+⌊x⌋+1x+1\lim _ { x \rightarrow 1 ^ { + } } \frac { \lfloor x \rfloor + 1 } { x + 1 }limx→1+​x+1⌊x⌋+1​ is

The limit lim⁡x→−∞x2+1x+1\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }limx→−∞​x+1x2+1​​ is

Let f(x)=3x+1−21−xf ( x ) = \frac { \sqrt { 3 x + 1 } - 2 } { 1 - x }f(x)=1−x3x+1​−2​ Which of the following is true for lim⁡x→1f(x)\lim _ { x \rightarrow 1 } f ( x )limx→1​f(x) ?

The limit lim⁡x→3+⌊x⌋−3x−3\lim _ { x \rightarrow 3 ^ { + } } \frac { \lfloor x \rfloor - 3 } { x - 3 }limx→3+​x−3⌊x⌋−3​ is

Let f(x)=3x2−5f ( x ) = 3 x ^ { 2 } - 5f(x)=3x2−5 Which of the following is equal to lim⁡h→0f(1+h)−f(1)h\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }limh→0​hf(1+h)−f(1)​ ?

Let f(x)=x2−3x−42x2−x−3f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 2 x ^ { 2 } - x - 3 }f(x)=2x2−x−3x2−3x−4​ Which of the following is true for lim⁡x→−1f(x)\lim _ { x \rightarrow - 1 } f ( x )limx→−1​f(x) ?

The largest set on which f(x)=x+1x2−1f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }f(x)=x2−1x+1​ is continuous is

The limit is lim⁡x→1x2+2x−3x−1\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } + 2 x - 3 } { x - 1 }limx→1​x−1x2+2x−3​ ?

The largest set on which f(x)=5x+35x2+3f ( x ) = \frac { 5 x + 3 } { 5 x ^ { 2 } + 3 }f(x)=5x2+35x+3​ is continuous is

The limit lim⁡x→0−ln⁡(ex−1)\lim _ { x \rightarrow 0 ^ { - } } \ln \left( e ^ { x } - 1 \right)limx→0−​ln(ex−1) is

Let f(x)=4−x−2x2+xf ( x ) = \frac { \sqrt { 4 - x } - 2 } { x ^ { 2 } + x }f(x)=x2+x4−x​−2​ Which of the following is true?

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) ?

Let f(x)=3x+4−11+xf ( x ) = \frac { \frac { 3 } { x + 4 } - 1 } { 1 + x }f(x)=1+xx+43​−1​ Which of the following is true for lim⁡x→−1f(x)\lim _ { x \rightarrow - 1 } f ( x )limx→−1​f(x) ?

The limit lim⁡x→2−⌊x⌋−2x−2\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor - 2 } { x - 2 }limx→2−​x−2⌊x⌋−2​ is

The limit lim⁡x→−∞x2+1x+1\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }limx→−∞​x+1x2+1​​ is

Let f(x)={x2+5xx<−10x=−1x−3x>−1f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 5 x & x < - 1 \\0 & x = - 1 \\x - 3 & x > - 1\end{array} \right.f(x)=⎩⎨⎧​x2+5x0x−3​x<−1x=−1x>−1​ Then ƒ is continuous at -1 if ƒ(-1) is redefined as

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = x4- 2x and y = 1 in the interval (-1, 0). ​

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

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

The limit $\lim _ { x \rightarrow 1 ^ { + } } \frac { \lfloor x \rfloor + 1 } { x + 1 }$ is

The limit $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }$ is

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

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

Let $f ( x ) = 3 x ^ { 2 } - 5$ Which of the following is equal to $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$ ?

Let $f ( x ) = \frac { x ^ { 2 } - 3 x - 4 } { 2 x ^ { 2 } - x - 3 }$ Which of the following is true for $\lim _ { x \rightarrow - 1 } f ( x )$ ?

The largest set on which $f ( x ) = \frac { x + 1 } { x ^ { 2 } - 1 }$ is continuous is

The limit is $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } + 2 x - 3 } { x - 1 }$ ?

The largest set on which $f ( x ) = \frac { 5 x + 3 } { 5 x ^ { 2 } + 3 }$ is continuous is

The limit $\lim _ { x \rightarrow 0 ^ { - } } \ln \left( e ^ { x } - 1 \right)$ is

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

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

Let $f ( x ) = \frac { \frac { 3 } { x + 4 } - 1 } { 1 + x }$ Which of the following is true for $\lim _ { x \rightarrow - 1 } f ( x )$ ?

The limit $\lim _ { x \rightarrow 2 ^ { - } } \frac { \lfloor x \rfloor - 2 } { x - 2 }$ is

The limit $\lim _ { x \rightarrow - \infty } \frac { \sqrt { x ^ { 2 } + 1 } } { x + 1 }$ is

Let $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } + 5 x & x < - 1 \\0 & x = - 1 \\x - 3 & x > - 1\end{array} \right.$ Then ƒ is continuous at -1 if ƒ(-1) is redefined as

Use the Intermediate Value Theorem to approximate the point of intersection of the function g(x) = x⁴- 2x and y = 1 in the interval (-1, 0).

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