8th Edition

ISBN: 9781305266636

Author: James Stewart

Publisher: Cengage Learning

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When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Textbook Question

Chapter 1.5, Problem 12E

Sketch the graph of the function and use it to determine the values of a for which lim_x_→a f(x) exists.

12. f ( x ) = { 1 + sin x if x < 0 cos x if 0 ≤ x ≤ π sin x if x > π

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Chapter 1.5, Problem 11E

Chapter 1.5, Problem 13E

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Chapter 1 Solutions

Single Variable Calculus

Ch. 1.1 - If f(x)=x+2x and g(u)=u+2u, is it true that f = g?Ch. 1.1 - If f(x)=x2xx1andg(x)=x is it true that f = g?Ch. 1.1 - The graph of a function f is given. (a) State the...Ch. 1.1 - The graphs of f and g are given. (a) State the...Ch. 1.1 - Figure 1 was recorded by an instrument operated by...Ch. 1.1 - Prob. 6E Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Determine whether the curve is the graph of a...Ch. 1.1 - Prob. 10E

Ch. 1.1 - Shown is a graph of the global average temperature...Ch. 1.1 - Trees grow faster and form wider rings in warm...Ch. 1.1 - You put some ice cubes in a glass, fill the glass...Ch. 1.1 - Three runners compete in a 100-meter race. 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S(t)...Ch. 1.3 - Prob. 52E 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 - Prob. 56E Ch. 1.3 - A ship is moving at a speed of 30 km/h parallel to...Ch. 1.3 - Prob. 58E 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 - Prob. 62E Ch. 1.3 - Prob. 63E Ch. 1.3 - Prob. 64E Ch. 1.3 - Prob. 65E Ch. 1.3 - Suppose g is an odd function and let h = f g. 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(a) Sketch the graph of...Ch. 1.6 - If f(x) = x + x, show that limx2 f(x) exists but...Ch. 1.6 - Prob. 56E Ch. 1.6 - If p is a polynomial, show that limxa p(x) = p(a).Ch. 1.6 - If r is a rational function, use Exercise 57 to...Ch. 1.6 - If limx1f(x)8x1=10, find limx1f(x).Ch. 1.6 - Prob. 60E Ch. 1.6 - Prob. 61E Ch. 1.6 - Prob. 62E Ch. 1.6 - Show by means of an example that limxa [f(x) g(x)]...Ch. 1.6 - Prob. 64E Ch. 1.6 - Is there a number a such that limx23x2+ax+a+3x2+x2...Ch. 1.6 - Prob. 66E Ch. 1.7 - Use the given graph of f to find a number such...Ch. 1.7 - Prob. 2E Ch. 1.7 - Use the given graph of f(x)=x to find a number ...Ch. 1.7 - Use the given graph of f(x) = x2 to find a number ...Ch. 1.7 - Prob. 5E Ch. 1.7 - Prob. 6E Ch. 1.7 - For the limit limx2(x33x+4)=6 illustrate...Ch. 1.7 - Prob. 8E Ch. 1.7 - (a) Use a graph to find a number such that...Ch. 1.7 - Given that limxcsc2x=, illustrate Definition 6 by...Ch. 1.7 - Prob. 11E Ch. 1.7 - A crystal growth furnace is used in research to...Ch. 1.7 - Prob. 13E Ch. 1.7 - Given that limx2 (5x 7) = 3, illustrate...Ch. 1.7 - Prob. 15E Ch. 1.7 - Prob. 16E Ch. 1.7 - Prob. 17E Ch. 1.7 - Prob. 18E Ch. 1.7 - Prove the statement using the , definition of a...Ch. 1.7 - Prove the statement using the , definition of a...Ch. 1.7 - Prob. 21E Ch. 1.7 - Prob. 22E Ch. 1.7 - Prob. 23E Ch. 1.7 - Prob. 24E Ch. 1.7 - Prob. 25E Ch. 1.7 - Prob. 26E Ch. 1.7 - Prob. 27E Ch. 1.7 - Prob. 28E Ch. 1.7 - Prob. 29E Ch. 1.7 - Prob. 30E Ch. 1.7 - Prob. 31E Ch. 1.7 - Prob. 32E Ch. 1.7 - Prob. 33E Ch. 1.7 - Prob. 34E Ch. 1.7 - Prob. 36E Ch. 1.7 - Prove that limxax=a if a 0. [Hint:Usexa=xax+a.]Ch. 1.7 - Prob. 38E Ch. 1.7 - Prob. 39E Ch. 1.7 - Prob. 40E Ch. 1.7 - Prob. 41E Ch. 1.7 - Prob. 42E Ch. 1.7 - Prob. 43E Ch. 1.7 - Prob. 44E 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...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 - Prob. 6E Ch. 1.8 - Sketch the graph of a function f that is...Ch. 1.8 - Prob. 8E Ch. 1.8 - The toll T charged for driving on a certain...Ch. 1.8 - Prob. 10E Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Prob. 12E Ch. 1.8 - Prob. 13E Ch. 1.8 - Prob. 14E Ch. 1.8 - Use the definition of continuity and the...Ch. 1.8 - Prob. 16E Ch. 1.8 - Prob. 17E Ch. 1.8 - Prob. 18E Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Prob. 20E Ch. 1.8 - Explain why the function is discontinuous at the...Ch. 1.8 - Prob. 22E Ch. 1.8 - Prob. 23E Ch. 1.8 - How would you remove the discontinuity of f? 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(a)...Ch. 1 - Prob. 2RE Ch. 1 - Prob. 3RE Ch. 1 - Prob. 4RE Ch. 1 - Prob. 5RE Ch. 1 - Prob. 6RE Ch. 1 - Prob. 7RE Ch. 1 - Prob. 8RE Ch. 1 - Prob. 9RE Ch. 1 - The graph of f is given. Draw the graphs of the...Ch. 1 - Prob. 11RE Ch. 1 - Prob. 12RE Ch. 1 - Prob. 13RE Ch. 1 - Prob. 14RE Ch. 1 - Prob. 15RE Ch. 1 - Prob. 16RE Ch. 1 - Prob. 17RE Ch. 1 - Prob. 18RE Ch. 1 - Prob. 19RE Ch. 1 - Prob. 20RE 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 - Prob. 24RE Ch. 1 - Prob. 25RE Ch. 1 - Prob. 26RE Ch. 1 - Prob. 27RE Ch. 1 - Prob. 28RE Ch. 1 - Prob. 29RE Ch. 1 - Prob. 30RE Ch. 1 - Prob. 31RE Ch. 1 - Prob. 32RE Ch. 1 - Find the limit. 33. limu1u41u3+5u26u Ch. 1 - Find the limit. 34. limx3x+6xx33x2 Ch. 1 - Prob. 35RE Ch. 1 - Prob. 36RE Ch. 1 - Prob. 37RE Ch. 1 - Prob. 38RE Ch. 1 - If 2x 1 f(x) x2 for 0 x 3, find limx1 f(x).Ch. 1 - Prob. 40RE Ch. 1 - Prob. 41RE Ch. 1 - Prob. 42RE Ch. 1 - Prob. 43RE Ch. 1 - Prob. 44RE Ch. 1 - Prob. 45RE Ch. 1 - Prob. 46RE Ch. 1 - Show that the function is continuous on its...Ch. 1 - Prob. 48RE Ch. 1 - Prob. 49RE Ch. 1 - Prob. 50RE Ch. 1 - Prob. 51RE Ch. 1 - Prob. 52RE Ch. 1 - Prob. 1P Ch. 1 - Prob. 2P Ch. 1 - Prob. 3P Ch. 1 - Prob. 4P Ch. 1 - Prob. 5P Ch. 1 - Prob. 6P Ch. 1 - The notation max{a, b, } means the largest of the...Ch. 1 - Prob. 8P Ch. 1 - Prob. 9P Ch. 1 - Prob. 10P Ch. 1 - Prove that if n is a positive integer, then 7n 1...Ch. 1 - Prob. 12P Ch. 1 - Prob. 13P Ch. 1 - Prob. 14P Ch. 1 - Prob. 15P Ch. 1 - Find numbers a and b such that limx0ax+b2x=1.Ch. 1 - Prob. 17P Ch. 1 - Prob. 18P Ch. 1 - Evaluate the following limits, if they exist,...Ch. 1 - Prob. 20P Ch. 1 - Prob. 21P Ch. 1 - A fixed point of a function f is a number c in its...Ch. 1 - Prob. 23P Ch. 1 - (a) The figure shows an isosceles triangle ABC...Ch. 1 - Prob. 25P

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Sketch the graph of the function and use it to determine the values of a for which lim x → a f ( x ) exists. 12. f ( x ) = { 1 + sin x if x < 0 cos x if 0 ≤ x ≤ π sin x if x > π

Solution Summary: The author explains how to draw the graph of f(x) and determine the value of a from it.

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Sketch the graph of the function and use it to determine the values of a for which lim x → a f ( x ) exists. 12. f ( x ) = { 1 + sin x if x &lt; 0 cos x if 0 ≤ x ≤ π sin x if x &gt; π

Solution Summary: The author explains how to draw the graph of f(x) and determine the value of a from it.

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Chapter 1 Solutions

Sketch the graph of the function and use it to determine the values of a for which lim x → a f ( x ) exists. 12. f ( x ) = { 1 + sin x if x < 0 cos x if 0 ≤ x ≤ π sin x if x > π