Question 1

Prove that the limit is correct using the appropriate definition. Show all work.

Accepted Answer

The answer of Prove that the limit is correct using...

Question 2

Find all horizontal and vertical asymptotes of f (x).   For each vertical asymptote, determine whether   or   on either side of the vertical asymptote.

Accepted Answer

A)  horizontal asymptote at y = 1, vertical asymptotes at x = sin (±2),         
B)  horizontal asymptote at y = 1, vertical asymptotes at x = ±2,         
C)  horizontal asymptote at y = sin (1), vertical asymptotes at x = ±2, Limits from both sides of each vertical asymptote are undefined. 
D)  horizontal asymptote at y = sin (1), vertical asymptotes at x = ±2,         
A)  horizontal asymptote at y = 1, vertical asymptotes at x = sin (±2),         
B)  horizontal asymptote at y = 1, vertical asymptotes at x = ±2,         
C)  horizontal asymptote at y = sin (1), vertical asymptotes at x = ±2, Limits from both sides of each vertical asymptote are undefined. 
D)  horizontal asymptote at y = sin (1), vertical asymptotes at x = ±2,

Question 3

Suppose that a state's tax code states that tax liability is   % on the first   of taxable earnings and   % on the remainder. Find the constants a and b in the tax function T(x) that make the function T(x) continuous.

Accepted Answer

A)    and   
B)    and   
C)    and   
D)    and   
A)    and   
B)    and   
C)    and   
D)    and

Question 4

Use graphical and numerical evidence to determine if   exists. If so, state the limit.

Accepted Answer

A)  The limit exists and is   at   . 
B)  The limit exists and is   at   . 
C)  The limit does not exist; the function is increasing without bound at   . 
D)  The limit does not exist; the function is decreasing without bound at   . 
A)  The limit exists and is   at   . 
B)  The limit exists and is   at   . 
C)  The limit does not exist; the function is increasing without bound at   . 
D)  The limit does not exist; the function is decreasing without bound at   .

Question 5

For the function graphed below, identify   or state that the limit does not exist.

Accepted Answer

A)  0 
B)    
C)    
D)  does not exist 
A)  0 
B)    
C)    
D)  does not exist

Question 6

Determine the limit.   Answer with a number,   ,   or that the limit does not exist.

Accepted Answer

A)    
B)  0 
C)    
D)    
E)  The limit does not exist. 
A)    
B)  0 
C)    
D)    
E)  The limit does not exist.

Question 7

Explain why the function fails to be continuous at   by indicating which of the conditions in the definition of continuity are not met.

Accepted Answer

A)    exists but   does not exist 
B)    exists but   does not exist 
C)  neither   nor   exist 
D)    exists and   exists but   
A)    exists but   does not exist 
B)    exists but   does not exist 
C)  neither   nor   exist 
D)    exists and   exists but

Question 8

Evaluate   where   .

Accepted Answer

A)    
B)    
C)    
D)  does not exist 
A)    
B)    
C)    
D)  does not exist

Question 9

Use the graph to identify all discontinuities f.

Accepted Answer

The functi

Question 10

Evaluate the limit, if it exists.

Accepted Answer

A)    
B)    
C)    
D)  does not exist 
A)    
B)    
C)    
D)  does not exist

Question 11

Find all horizontal and vertical asymptotes of f(x).   For each vertical asymptote, determine whether   or   on either side of the vertical asymptote.

Accepted Answer

A)  horizontal asymptotes at   ; there are no vertical asymptotes. 
B)  horizontal asymptote at   , vertical asymptote at   ;   on both sides of   
C)  horizontal asymptote at   , vertical asymptote at   ;   on both sides of   
D)  horizontal asymptotes at   , vertical asymptote at   ;   and   
A)  horizontal asymptotes at   ; there are no vertical asymptotes. 
B)  horizontal asymptote at   , vertical asymptote at   ;   on both sides of   
C)  horizontal asymptote at   , vertical asymptote at   ;   on both sides of   
D)  horizontal asymptotes at   , vertical asymptote at   ;   and

Question 12

Suppose the length of an animal t days after birth is given by h(t).   mm What is the eventual length of the animal (i.e., h(t) as   )?

Accepted Answer

A)  0 mm 
B)    mm 
C)    mm 
D)    mm 
A)  0 mm 
B)    mm 
C)    mm 
D)    mm

Question 13

Find   in terms of   for   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 14

Evaluate the limit, if it exists.

Accepted Answer

A)    
B)    
C)    
D)  does not exist 
A)    
B)    
C)    
D)  does not exist

Question 15

Evaluate the limit, if it exists.

Accepted Answer

A)  0 
B)    
C)    
D)  does not exist 
A)  0 
B)    
C)    
D)  does not exist

Question 16

Find   in terms of   for   .

Accepted Answer

A)    
B)    
C)    
D)    
A)    
B)    
C)    
D)

Question 17

Determine if f is continuous at   from the right.

Accepted Answer

A)    but   is continuous from the right 
B)    so   is continuous from the right 
C)    so   is not continuous from the right 
D)    but   is not continuous from the right 
A)    but   is continuous from the right 
B)    so   is continuous from the right 
C)    so   is not continuous from the right 
D)    but   is not continuous from the right

Question 18

Determine if f is continuous at   from the right.

Accepted Answer

A)    , but   is continuous from the right 
B)    , so   is continuous from the right 
C)    , so   is not continuous from the right 
D)    , but   is not continuous from the right 
A)    , but   is continuous from the right 
B)    , so   is continuous from the right 
C)    , so   is not continuous from the right 
D)    , but   is not continuous from the right

Question 19

Evaluate the limit, if it exists.

Accepted Answer

A)    
B)    
C)    
D)  does not exist 
A)    
B)    
C)    
D)  does not exist

Question 20

Complete the tables appropriately and use the numerical evidence to conjecture the value of   . x   2.9
2)99
2)999
2)9999
X   3.1
3)01
3)001
3)0001

Accepted Answer

A)  0 
B)  3 
C)  -3 
D)  -81 
A)  0 
B)  3 
C)  -3 
D)  -81

Prove that the limit is correct using the appropriate definition. Show all work.

Find all horizontal and vertical asymptotes of f (x). For each vertical asymptote, determine whether or on either side of the vertical asymptote.

Suppose that a state's tax code states that tax liability is % on the first of taxable earnings and % on the remainder. Find the constants a and b in the tax function T(x) that make the function T(x) continuous.

Use graphical and numerical evidence to determine if exists. If so, state the limit.

For the function graphed below, identify or state that the limit does not exist.

Determine the limit. Answer with a number, , or that the limit does not exist.

Explain why the function fails to be continuous at by indicating which of the conditions in the definition of continuity are not met.

Evaluate where .

Use the graph to identify all discontinuities f.

Evaluate the limit, if it exists.

Find all horizontal and vertical asymptotes of f(x). For each vertical asymptote, determine whether or on either side of the vertical asymptote.

Suppose the length of an animal t days after birth is given by h(t). mm What is the eventual length of the animal (i.e., h(t) as )?

Find in terms of for .

Evaluate the limit, if it exists.

Evaluate the limit, if it exists.

Find in terms of for .

Determine if f is continuous at from the right.

Determine if f is continuous at from the right.

Evaluate the limit, if it exists.

Complete the tables appropriately and use the numerical evidence to conjecture the value of . x 2.9 2)99 2)999 2)9999 X 3.1 3)01 3)001 3)0001

Preliminaries

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters

Exam 2: Limits and Continuity

Prove that the limit is correct using the appropriate definition. Show all work.

Find all horizontal and vertical asymptotes of f (x). For each vertical asymptote, determine whether or on either side of the vertical asymptote.

Suppose that a state's tax code states that tax liability is % on the first of taxable earnings and % on the remainder. Find the constants a and b in the tax function T(x) that make the function T(x) continuous.

Use graphical and numerical evidence to determine if exists. If so, state the limit.

For the function graphed below, identify or state that the limit does not exist.

Determine the limit. Answer with a number, , or that the limit does not exist.

Explain why the function fails to be continuous at by indicating which of the conditions in the definition of continuity are not met.

Evaluate where .

Use the graph to identify all discontinuities f.

Evaluate the limit, if it exists.

Find all horizontal and vertical asymptotes of f(x). For each vertical asymptote, determine whether or on either side of the vertical asymptote.

Suppose the length of an animal t days after birth is given by h(t). mm What is the eventual length of the animal (i.e., h(t) as )?

Find in terms of for .

Evaluate the limit, if it exists.

Evaluate the limit, if it exists.

Find in terms of for .

Determine if f is continuous at from the right.

Determine if f is continuous at from the right.

Evaluate the limit, if it exists.

Complete the tables appropriately and use the numerical evidence to conjecture the value of . x 2.9 2)99 2)999 2)9999 X 3.1 3)01 3)001 3)0001

Preliminaries

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Vector Calculus

Second Order Differential Equations

Filters