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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781305266636

Chapter 1.7, Problem 24E

Textbook Problem

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Prove the statement using the *ε*, *δ* definition of a limit.

**24.**

To determine

**To prove:** The limit of a function *c* by using the

**Definition used:**

“Let *f* be a function defined on some open interval that contains the number *a*, except possibly at *a* itself. Then, the limit of *x* approaches *a* is *L*,

**To guess:** The number

Let

By the definition of

Therefore, find a number

Single Variable Calculus

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In...Ch. 1.8 - How would you remove the discontinuity of f? 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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 make 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 (f ...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...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,...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) = |...Ch. 1.8 - A Tibetan monk leaves the monastery at 7:00 am and...Ch. 1 - (a) What is a function? What are its domain and...Ch. 1 - Discuss four ways of representing a function....Ch. 1 - (a) What is an even function? How can you tell if...Ch. 1 - What is an increasing function?Ch. 1 - What is a mathematical model?Ch. 1 - Give an example of each type of function. (a)...Ch. 1 - Sketch by hand, on the same axes, the graphs of...Ch. 1 - Draw, by hand, a rough sketch of the graph of each...Ch. 1 - Suppose that f has domain A and g has domain B....Ch. 1 - How is the composite function f g defined? What...Ch. 1 - Suppose the graph of f is given. Write an equation...Ch. 1 - Explain what each of the following means and...Ch. 1 - Describe several ways in which a limit can fail to...Ch. 1 - What does it mean to say that the line x = a is a...Ch. 1 - State the following Limit Laws. (a) Sum Law (b)...Ch. 1 - What does the Squeeze Theorem say?Ch. 1 - (a) What does it mean for f to be continuous at a?...Ch. 1 - (a) Give examples of functions that are continuous...Ch. 1 - What does the Intermediate Value Theorem say?Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Determine whether the statement is true or false....Ch. 1 - Let f be the function whose graph is given. (a)...Ch. 1 - Determine whether each curve is the graph of a...Ch. 1 - If f(x) = x2 2x + 3, evaluate the difference...Ch. 1 - Sketch a rough graph of the yield of a crop as a...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Find the domain and range of the function. Write...Ch. 1 - Suppose that the graph of f is given. Describe how...Ch. 1 - The graph of f is given. Draw the graphs of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Use transformations to sketch the graph of the...Ch. 1 - Determine whether f is even, odd, or neither even...Ch. 1 - Find an expression for the function whose graph...Ch. 1 - If f(x)=x and g(x) = sin x, find the functions (a)...Ch. 1 - Express the function F(x)=1/x+x as a composition...Ch. 1 - A small-appliance manufacturer finds that it costs...Ch. 1 - The graph of f is given. (a) Find each limit, or...Ch. 1 - Sketch the graph of an example of a function f...Ch. 1 - Find the limit. 25. limx0cos(x+sinx)Ch. 1 - Find the limit. 26. limx3x29x2+2x3Ch. 1 - Find the limit. 27. limx3x29x2+2x3Ch. 1 - Find the limit. 28. limx1+x29x2+2x3Ch. 1 - Find the limit. 29. limh0(h1)3+1hCh. 1 - Find the limit. 30. limt2t24t38Ch. 1 - Find the limit. 31. limr9r(r9)4Ch. 1 - Find the limit. 32. limv4+4v4vCh. 1 - Find the limit. 33. limu1u41u3+5u26uCh. 1 - Find the limit. 34. limx3x+6xx33x2Ch. 1 - Find the limit. 35. lims164ss16Ch. 1 - Find the limit. 36. limv2v2+2v8v416Ch. 1 - Find the limit. 37. limx011x2xCh. 1 - Find the limit. 38. limx1(1x1+1x23x+2)Ch. 1 - If 2x 1 f(x) x2 for 0 x 3, find limx1 f(x).Ch. 1 - Prove that limx0 x2 cos(1/x2) = 0.Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Prove the statement using the precise definition...Ch. 1 - Let f(x)={xifx03xif0x3(x3)2ifx3 (a) Evaluate each...Ch. 1 - Let g(x)={2xx2if0x22xif2x3x4if3x4ifx4 (a) For each...Ch. 1 - Show that the function is continuous on its...Ch. 1 - Show that the function is continuous on its...Ch. 1 - Use the Intermediate Value Theorem to show that...Ch. 1 - Use the Intermediate Value Theorem to show that...Ch. 1 - Suppose that |f(x)| g(x) for all x, where limxa...Ch. 1 - Let f(x) = x + x. (a) For what values of a does...Ch. 1 - Solve the equation |2x 1| |x + 5| = 3.Ch. 1 - Solve the inequality |x 1| |x 3 | 5.Ch. 1 - Sketch the graph of the function f(x) = |x2 4|x|...Ch. 1 - Sketch the graph of the function g(x) = |x2 1| ...Ch. 1 - Draw the graph of the equation x + |x| = y + |y|.Ch. 1 - Sketch the region in the plane consisting of all...Ch. 1 - The notation max{a, b, } means the largest of the...Ch. 1 - Sketch the region in the plane defined by each of...Ch. 1 - A driver sets out on a journey. For the first half...Ch. 1 - Is it true that f (g + h) = f g + f h?Ch. 1 - Prove that if n is a positive integer, then 7n 1...Ch. 1 - Prove that 1 + 3 + 5+ ...+ (2n 1)=n2.Ch. 1 - If f0(x) = x2 and fn+1(x) = f0(fn(x)) for n = 0,...Ch. 1 - (a) If f0(x)=12x and fn+1 = f0 fn for n = 0, 1,...Ch. 1 - Evaluate limx1x31x1.Ch. 1 - Find numbers a and b such that limx0ax+b2x=1.Ch. 1 - Evaluate limx02x12x+1x.Ch. 1 - The figure shows a point P on the parabola y = x2...Ch. 1 - Evaluate the following limits, if they exist,...Ch. 1 - Sketch the region in the plane defined by each of...Ch. 1 - Find all values of a such that f is continuous on...Ch. 1 - A fixed point of a function f is a number c in its...Ch. 1 - If limxa [f(x) + g(x)] = 2 and limxa [f(x) g(x)]...Ch. 1 - (a) The figure shows an isosceles triangle ABC...Ch. 1 - (a) If we start from 0 latitude and proceed in a...

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