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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 1.6, Problem 59E

Textbook Problem

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If

To determine

**To find**:

Here, the limit exists therefore, rewrite using the limit laws.

Calculus (MindTap Course List)

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Write equations...Ch. 1.3 - Explain how each graph is obtained from the graph...Ch. 1.3 - The graph of y=f(x) is given. Match each equation...Ch. 1.3 - The graph of f is given. Draw the graphs of the...Ch. 1.3 - The graph of f is given. Use it to graph the...Ch. 1.3 - The graph of y=3xx2 is given. Use transformations...Ch. 1.3 - The graph of y=3xx2 is given. 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H(x)=2+|x|8Ch. 1.3 - Express the function in the form fgh....Ch. 1.3 - Use the table to evaluate each expression. a...Ch. 1.3 - Use the given graphs of f and g to evaluate each...Ch. 1.3 - Use the given graphs of f and g to estimate the...Ch. 1.3 - A stone is dropped into a lake, creating a...Ch. 1.3 - A spherical balloon is being inflated and the...Ch. 1.3 - A ship is moving at a speed of 30 km/h parallel to...Ch. 1.3 - An airplane is flying at a speed of 350 mi/h at an...Ch. 1.3 - The Heaviside function H is defined by...Ch. 1.3 - The Heaviside function defined in Exercise 59 can...Ch. 1.3 - Let f and g be linear functions with equations...Ch. 1.3 - If you invest x dollars at 4 interest compounded...Ch. 1.3 - a If g(x)=2x+1 and h(x)=4x2+4x+7, find a function...Ch. 1.3 - If f(x)=x+4 and h(x)=4x1, find a function g such...Ch. 1.3 - Suppose g is an even function and let h=fg. Is h...Ch. 1.3 - Suppose g is an odd function and let h=fg. 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[Hint:Usexa=|xa|x+a.]Ch. 1.7 - If H is the Heaviside function defined in Example...Ch. 1.7 - If the function f is defined by...Ch. 1.7 - By comparing Definitions 2, 3, and 4, prove...Ch. 1.7 - How close to 3 do we have to take x so that...Ch. 1.7 - Prove, using Definition 6, that limx31(x+3)4=.Ch. 1.7 - Prove that limx15(x+1)3=Ch. 1.7 - Suppose that limxaf(x)= and limxag(x)=c, where c...Ch. 1.8 - Write an equation that expresses the fact that a...Ch. 1.8 - If f is continuous on (,), what can you say about...Ch. 1.8 - a From the graph of f, state the numbers at which...Ch. 1.8 - From the graph of g, state the intervals on which...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - The toll T charged for driving on a certain...Ch. 1.8 - Explain why each function is continuous or...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - How would you remove the discontinuity of f? In...Ch. 1.8 - How would you remove the discontinuity of f? In...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Explain, using Theorems 4, 5, 7, and 9, why the...Ch. 1.8 - Locate the discontinuities of the function and...Ch. 1.8 - Locate the discontinuities of the function and...Ch. 1.8 - Use continuity to evaluate the limit. limx2x20x2Ch. 1.8 - Use continuity to evaluate the limit....Ch. 1.8 - Use continuity to evaluate the limit. limx/4x2tanxCh. 1.8 - Use continuity to evaluate the limit....Ch. 1.8 - Show that f is continuous on (,)....Ch. 1.8 - Show that f is continuous on (,)....Ch. 1.8 - Find the numbers at which f is discontinuous. At...Ch. 1.8 - Find the numbers at which f is discontinuous. At...Ch. 1.8 - Find the numbers at which f is discontinuous. At...Ch. 1.8 - The gravitational force exerted by the planet...Ch. 1.8 - For what value of the constant c is the function f...Ch. 1.8 - Find the values of a and b that makes f continuous...Ch. 1.8 - Suppose f and g are continuous functions such that...Ch. 1.8 - Let f(x)=1/x and g(x)=1/x2. a Find (fg)(x) b Is fg...Ch. 1.8 - Which of the following functions f has a removable...Ch. 1.8 - Suppose that a function f is continuous on [0,1]...Ch. 1.8 - If f(x)=x2+10 sin x, show that there is a number c...Ch. 1.8 - Suppose f is continuous on [1,5] and the only...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - Use the Intermediate Value Theorem to show that...Ch. 1.8 - a Prove that the equation has at least one real...Ch. 1.8 - a Prove that the equation has at least one real...Ch. 1.8 - a Prove that the equation has at least one real...Ch. 1.8 - a Prove that the equation has at least one real...Ch. 1.8 - Prove, without graphing, that the graph of the...Ch. 1.8 - Prove, without graphing, that the graph of the...Ch. 1.8 - Prove that f is continuous at a if and only if...Ch. 1.8 - To prove that sine is continuous, we need to show...Ch. 1.8 - Prove that cosine is a continuous function.Ch. 1.8 - a Prove Theorem 4, part 3. b Prove Theorem 4, part...Ch. 1.8 - For what values of x is f continuous?...Ch. 1.8 - For what values of x is g continuous?...Ch. 1.8 - Is there a number that is exactly 1 more than its...Ch. 1.8 - If a and b are positive numbers, prove that the...Ch. 1.8 - Show that the function f(x)={x4sin(1/x)ifx00ifx=0...Ch. 1.8 - a Show that the absolute value function F(x)=|x|...Ch. 1.8 - A Tibetan monk leaves the monastery at 7.00 AMand...Ch. 1.R - a What is a function? What are its domain and...Ch. 1.R - Discuss four ways of representing a function....Ch. 1.R - a What is an even function? How can you tell if a...Ch. 1.R - What is an increasing function?Ch. 1.R - What is a mathematical model?Ch. 1.R - Give an example of each type of function. a Linear...Ch. 1.R - Sketch by hand, on the same axes, the graphs of...Ch. 1.R - Draw, by hand, a rough sketch of the graph of each...Ch. 1.R - Suppose that f has domain A and g has domain B. a...Ch. 1.R - How is the composite function fg defined? What is...Ch. 1.R - Suppose the graph of f is given. Write an equation...Ch. 1.R - Explain what each of the following means and...Ch. 1.R - Describe several ways in which a limit can fail to...Ch. 1.R - What does it mean to say that the line x=a is a...Ch. 1.R - State the following Limit Laws. a Sum Law b...Ch. 1.R - What does the Squeeze Theorem say?Ch. 1.R - a What does it mean for f to be continuous at a? b...Ch. 1.R - a Give examples of functions that are continuous...Ch. 1.R - What does the Intermediate Value Theorem say?Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. 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If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Determine whether statement is true or false. If...Ch. 1.R - Let f be the function whose graph is given, a...Ch. 1.R - Determine whether each curve is the graph of a...Ch. 1.R - If f(x)=x22x+3, evaluate the different quotient...Ch. 1.R - Sketch a rough graph of the yield of a crop as a...Ch. 1.R - Find the domain and range of the function. Write...Ch. 1.R - Find the domain and range of the function. Write...Ch. 1.R - Find the domain and range of the function. Write...Ch. 1.R - Find the domain and range of the function. Write...Ch. 1.R - Suppose that the graph of f is given. Describe how...Ch. 1.R - The graph of f is given. Draw the graphs of the...Ch. 1.R - Use transformations to sketch the graph of the...Ch. 1.R - Use transformations to sketch the graph of the...Ch. 1.R - Use transformations to sketch the graph of the...Ch. 1.R - Use transformations to sketch the graph of the...Ch. 1.R - Use transformations to sketch the graph of the...Ch. 1.R - Use transformations to sketch the graph of the...Ch. 1.R - Determine whether f is even, odd, or neither even...Ch. 1.R - Find an expression for the function whose graph...Ch. 1.R - If f(x)=x and g(x)=sinx, find the functions a fg,...Ch. 1.R - Express the function F(x)=1x+x as a composition of...Ch. 1.R - Life expectancy improved dramatically in the 20th...Ch. 1.R - A small-appliance manufacturer finds that it costs...Ch. 1.R - The graph of f is given, a Find each limit, or...Ch. 1.R - Sketch the graph of an example of a function f...Ch. 1.R - Find the limit. limx0cos(x+sinx)Ch. 1.R - Find the limit. limx3x29x2+2x3Ch. 1.R - Find the limit. limx3x29x2+2x3Ch. 1.R - Find the limit. limx1+x29x2+2x3Ch. 1.R - Find the limit. limh0(h1)3+1hCh. 1.R - Find the limit. limt2t24t38Ch. 1.R - Find the limit. limr9r(r9)4Ch. 1.R - Find the limit. limv4+4v|4v|Ch. 1.R - Find the limit. limu1u41u3+5u26uCh. 1.R - Find the limit. limx3x+6xx33x2Ch. 1.R - Find the limit. lims164ss16Ch. 1.R - Find the limit. limv2v2+2v8v416Ch. 1.R - Find the limit. limx011x2xCh. 1.R - Find the limit. limx1(1x1+1x23x+2)Ch. 1.R - If 2x1f(x)x2 for 0x3, find limx1f(x).Ch. 1.R - Prove that limx0x2cos(1/x2)=0.Ch. 1.R - Prove the statement using the precise definition...Ch. 1.R - Prove the statement using the precise definition...Ch. 1.R - Prove the statement using the precise definition...Ch. 1.R - Prove the statement using the precise definition...Ch. 1.R - Let f(x)={xifx03xif0x3(x3)2ifx3 a Evaluate each...Ch. 1.R - Let g(x)={2xx2if0x22xif2x3x4if3x4ifx4 a For each...Ch. 1.R - Show that the function is continuous on its...Ch. 1.R - Show that the function is continuous on its...Ch. 1.R - Use the Intermediate Value Theorem to show that...Ch. 1.R - Use the Intermediate Value Theorem to show that...Ch. 1.R - Suppose that |f(x)|g(x) for all x, where...Ch. 1.R - Let f(x)=x+x. a For what values of a does...Ch. 1.PPS - Solve the equation |2x1||x+5|=3.Ch. 1.PPS - Solve the inequality |x1||x3|5.Ch. 1.PPS - Sketch the graph of the function f(x)=|x24|x|+3|.Ch. 1.PPS - Sketch the graph of the function g(x)=|x21||x24|.Ch. 1.PPS - Draw the graph of the equation x+|x|=y+|y|.Ch. 1.PPS - Sketch the region in the plane consisting of all...Ch. 1.PPS - The notation max {a,b,...} means the largest of...Ch. 1.PPS - Sketch the region in the plane defined by each of...Ch. 1.PPS - A driver sets out on a journey. For the first half...Ch. 1.PPS - Is it true that f(g+h)=fg+fh?Ch. 1.PPS - Prove that if n is a positive integer, then 7n1 is...Ch. 1.PPS - Prove that 1+3+5+...+(2n1)=n2.Ch. 1.PPS - If f0(x)=x2 and fn+1(x)=f0(fn(x)) for n=0,1,2,...,...Ch. 1.PPS - a If f0(x)=12x and fn+1=f0fn for n=0,1,2,....,...Ch. 1.PPS - Evaluate limx1x31x1.Ch. 1.PPS - Find numbers a and b such that limx0ax+b2x=1.Ch. 1.PPS - Evaluate limx0|2x1||2x+1|x.Ch. 1.PPS - The figure shows a point P on the parabola y=x2...Ch. 1.PPS - Evaluate the following limits, if they exists,...Ch. 1.PPS - Sketch the region in the plane defined by each of...Ch. 1.PPS - Find all values of a such that f is continuous on...Ch. 1.PPS - A fixed point of a function f is a number c in its...Ch. 1.PPS - If limxa[f(x)+g(x)]=2 and limxa[f(x)g(x)]=1, find...Ch. 1.PPS - a The figure shows an isosceles triangle ABC with...Ch. 1.PPS - a If we start from 0 latitude and proceed in a...

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