Question 1

Determine the following limit. (Hint: Use the graph to calculate the limit.)

Accepted Answer

A) 6 
B) 5 
C) 1 
D) 4 
E) does not exist 
A) 6 
B) 5 
C) 1 
D) 4 
E) does not exist

Question 2

Find the value of c guaranteed by the Intermediate Value Theorem.   .

Accepted Answer

A) 3 
B) 2 
C) 5 
D) 7 
E) 6 
A) 3 
B) 2 
C) 5 
D) 7 
E) 6

Question 3

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution.A cyclist is riding on a path whose elevation is modeled by the function   where x and   are measured in miles. Find the rate of change of elevation when x = 4.0.

Accepted Answer

A) calculus , 1.28 
B) precalculus , 1.28 
C) calculus , 0.16 
D) precalculus , 0.16 
E) precalculus , 0.41 
A) calculus , 1.28 
B) precalculus , 1.28 
C) calculus , 0.16 
D) precalculus , 0.16 
E) precalculus , 0.41

Question 4

Find all vertical asymptotes (if any) of the function   .

Accepted Answer

A) x = 4, 3, -6 
B) x = 3 
C) x = -4, -3 
D) x = -3 
E) x = 4, 3 
A) x = 4, 3, -6 
B) x = 3 
C) x = -4, -3 
D) x = -3 
E) x = 4, 3

Question 5

Find the limit (if it exists). Note that   represents the greatest integer function.

Accepted Answer

A) 17 
B) 15 
C) 0 
D) 16 
E) Limit does not exist. 
A) 17 
B) 15 
C) 0 
D) 16 
E) Limit does not exist.

Question 6

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution.Find the distance traveled in 17 seconds by an object moving with a velocity of   feet per second.

Accepted Answer

A) precalculus , 168.0772 ft 
B) calculus , 169.4497 ft 
C) calculus , 168.0772 ft 
D) precalculus , 169.4497 ft 
E) precalculus , 170.7997 ft 
A) precalculus , 168.0772 ft 
B) calculus , 169.4497 ft 
C) calculus , 168.0772 ft 
D) precalculus , 169.4497 ft 
E) precalculus , 170.7997 ft

Question 7

Use a graphing utility to graph the function   and determine the one-sided limit   .

Accepted Answer

A)    
B)    
C) 0 
D) 12 
E) 8 
A)    
B)    
C) 0 
D) 12 
E) 8

Question 8

Find the x-values (if any) at which   is not continuous.

Accepted Answer

A)    is continuous for all real x. 
B)    is not continuous at   and both the discontinuities are nonremovable. 
C)    is not continuous only at   and   has a removable discontinuity at   . 
D)    is not continuous only at   and   has a removable discontinuity at   . 
E)    is not continuous at   and   has a removable discontinuity at   . 
A)    is continuous for all real x. 
B)    is not continuous at   and both the discontinuities are nonremovable. 
C)    is not continuous only at   and   has a removable discontinuity at   . 
D)    is not continuous only at   and   has a removable discontinuity at   . 
E)    is not continuous at   and   has a removable discontinuity at   .

Question 9

Determine the following limit. (Hint: Use the graph to calculate the limit.)

Accepted Answer

A) does not exist 
B) 2 
C) 4 
D) 8 
E) 0 
A) does not exist 
B) 2 
C) 4 
D) 8 
E) 0

Question 10

A 35-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of   where x is the distance between the base of the ladder and the house. Find the limit of r as   .

Accepted Answer

A)    
B) 70 
C) 0 
D) 35 
E)    
A)    
B) 70 
C) 0 
D) 35 
E)

Question 11

Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point.

Accepted Answer

A) does not exist 
B) 74 
C) -74 
D) -80 
E) 80 
A) does not exist 
B) 74 
C) -74 
D) -80 
E) 80

Question 12

Suppose that   and   . Find the following limit.

Accepted Answer

A)    
B) -63 
C)    
D) does not exist. 
E) 63 
A)    
B) -63 
C)    
D) does not exist. 
E) 63

Question 13

Find the x-values (if any) at which the function   is not continuous. Which of the discontinuities are removable?

Accepted Answer

A) continuous everywhere 
B)    , removable 
C)    , removable. 
D)    , not removable. 
E) x = 0, removable 
A) continuous everywhere 
B)    , removable 
C)    , removable. 
D)    , not removable. 
E) x = 0, removable

Question 14

Find the x-values (if any) at which the function   is not continuous. Which of the discontinuities are removable?

Accepted Answer

A) 3 and -3, not removable. 
B) continuous everywhere 
C) 3 and -3, removable. 
D) discontinuous everywhere 
E) 0, removable 
A) 3 and -3, not removable. 
B) continuous everywhere 
C) 3 and -3, removable. 
D) discontinuous everywhere 
E) 0, removable

Question 15

Find the vertical asymptotes (if any) of the function   .

Accepted Answer

A) x = 4 
B) x = -4 
C) x = 3 
D) x = -3 
E) x = 12 
A) x = 4 
B) x = -4 
C) x = 3 
D) x = -3 
E) x = 12

Question 16

Suppose that   . Find the following limit.

Accepted Answer

A) 26 
B) -2 
C) -13 
D) -13c 
E) -26 
A) 26 
B) -2 
C) -13 
D) -13c 
E) -26

Question 17

Suppose that   and   . Find the following limit.

Accepted Answer

A) 140 
B) -4 
C) 0 
D) -24 
E) -10 
A) 140 
B) -4 
C) 0 
D) -24 
E) -10

Question 18

Find constants a and b such that the function   is continuous on the entire real line.

Accepted Answer

A) a = 1 , b = 0 
B) a = 1 , b = 3 
C) a = 1 , b = -3 
D) a = -1 , b = -3 
E) a = -1 , b = 3 
A) a = 1 , b = 0 
B) a = 1 , b = 3 
C) a = 1 , b = -3 
D) a = -1 , b = -3 
E) a = -1 , b = 3

Question 19

Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution.Find the area of the shaded region bounded by the triangle with vertices (0,0), (7,8), (15,0).

Accepted Answer

A) precalculus , 60.0 
B) precalculus , 120.0 
C) calculus , 180.0 
D) calculus , 120.0 
E) precalculus , 180.0 
A) precalculus , 60.0 
B) precalculus , 120.0 
C) calculus , 180.0 
D) calculus , 120.0 
E) precalculus , 180.0

Question 20

Find the limit (if it exists).

Accepted Answer

A)    
B) 1 
C) 0 
D)    
E) Limit does not exist. 
A)    
B) 1 
C) 0 
D)    
E) Limit does not exist.

Determine the following limit. (Hint: Use the graph to calculate the limit.)

Find the value of c guaranteed by the Intermediate Value Theorem. .

Find all vertical asymptotes (if any) of the function .

Find the limit (if it exists). Note that represents the greatest integer function.

Use a graphing utility to graph the function and determine the one-sided limit .

Find the x-values (if any) at which is not continuous.

Determine the following limit. (Hint: Use the graph to calculate the limit.)

A 35-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of where x is the distance between the base of the ladder and the house. Find the limit of r as .

Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point.

Suppose that and . Find the following limit.

Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable?

Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable?

Find the vertical asymptotes (if any) of the function .

Suppose that . Find the following limit.

Suppose that and . Find the following limit.

Find constants a and b such that the function is continuous on the entire real line.

Find the limit (if it exists).

Prerequisites

Functions and Their Graphs

Polynomial and Rational Functions

Differentiation

Applications of Differentiation

Integration

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions and Calculus

Trigonometric Functions

Analytic Trigonometry

Trigonometric Functions and Calculus

Topics in Analytic Geometry

Additional Topics in Trigonometry

Filters

Exam 4: Limits and Their Properties

Determine the following limit. (Hint: Use the graph to calculate the limit.)

Find the value of c guaranteed by the Intermediate Value Theorem. .

Find all vertical asymptotes (if any) of the function .

Find the limit (if it exists). Note that represents the greatest integer function.

Use a graphing utility to graph the function and determine the one-sided limit .

Find the x-values (if any) at which is not continuous.

Determine the following limit. (Hint: Use the graph to calculate the limit.)

A 35-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of where x is the distance between the base of the ladder and the house. Find the limit of r as .

Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point.

Suppose that and . Find the following limit.

Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable?

Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable?

Find the vertical asymptotes (if any) of the function .

Suppose that . Find the following limit.

Suppose that and . Find the following limit.

Find constants a and b such that the function is continuous on the entire real line.

Find the limit (if it exists).

Prerequisites

Functions and Their Graphs

Polynomial and Rational Functions

Differentiation

Applications of Differentiation

Integration

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions and Calculus

Trigonometric Functions

Analytic Trigonometry

Trigonometric Functions and Calculus

Topics in Analytic Geometry

Additional Topics in Trigonometry

Filters