Explanation Definition used: Greatest integer function: 〚 x 〛 = the largest integer that is less than or equal to x . Theorem 1: The limit lim x → a f ( x ) = L if and only if lim x → a − f ( x ) = L = lim x → a + f ( x ) . Proof: Construct a table of y = 〚 x 〛 for some arbitrary values of x. x −3 −2.5 −2 −1.9 −1 −0.7 0 0.7 1 1.9 2 2.5 3 y −3 −3 −2 −2 −1 −1 0 0 1 1 2 2 3 Draw the graph of y = 〚 x 〛 using the table as shown in Figure 1. Construct a table of y = 〚 − x 〛 for some arbitrary values of x. x −3 −2.5 −2 −1.9 −1 −0.7 0 0.7 1 1.9 2 2.5 3 y 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 Draw the graph of y = 〚 − x 〛 using the table as shown in Figure 2. Obtain the limit of the function f ( x ) = 〚 x 〛 + 〚 − x 〛 as x approaches 2. lim x → 2 f ( x ) = lim x → 2 ( 〚 x 〛 + 〚 − x 〛 ) Obtain the limit of the function f ( x ) = 〚 x 〛 + 〚 − x 〛 as x approaches left-hand side of 2. lim x → 2 − f ( x ) = lim x → 2 − ( 〚 x 〛 + 〚 − x 〛 ) = lim x → 2 − 〚 x 〛 + lim x → 2 − 〚 − x 〛 [ by limit law 1 ] From the Figure 1 and 2, it is observed that lim x → 2 − 〚 x 〛 = 1 and lim x → 2 − 〚 − x 〛 = − 2 . lim x → 2 − 〚 x 〛 + lim x → 2 − 〚 − x 〛 = 1 + ( − 2 ) = 1 − 2 = − 1 Obtain the limit of the function f ( x ) = 〚 x 〛 + 〚 − x 〛 as x approaches right-hand side of 2. lim x → 2 + f ( x ) = lim x → 2 + ( 〚 x 〛 + 〚 − x 〛 ) = lim x → 2 + 〚 x 〛 + lim x → 2 + 〚 − x 〛 [ by limit law 1 ] From the Figure 1 and 2, it is observed that lim x → 2 + 〚 x 〛 = 2 and lim x → 2 + 〚 − x 〛 = − 3 . lim x → 2 + 〚 x 〛 + lim x → 2 + 〚 − x 〛 = 2 + ( − 3 ) = 2 − 3 = − 1 Therefore, the left and right side limits are equal

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 2.3, Problem 55E

To determine

To show: If f(x)=〚x〛+〚−x〛 , then limx→2f(x) exists but is not equal to f(2).

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Chapter 2.3, Problem 54E

Chapter 2.3, Problem 56E

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Chapter 2 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 2.1 - A Lank holds 1000 gallons o f water, which drains...Ch. 2.1 - A cardiac monitor is used to measure the heart...Ch. 2.1 - The point P(2, 1) lies on the curve y = 1/(1 x)....Ch. 2.1 - The point P(0.5, 0) lies on the curve y = cos x....Ch. 2.1 - If a ball is thrown into the air with a velocity...Ch. 2.1 - If a rock is thrown upward on the planet Mars with...Ch. 2.1 - The table shows the position of a motorcyclist...Ch. 2.1 - The displacement (in centimeters) of a particle...Ch. 2.1 - The point P(1, 0) lies on the curve y = sin(l0/x)....Ch. 2.2 - Prob. 1E

Ch. 2.2 - Explain what it means to say that...Ch. 2.2 - Explain the meaning of each of the following. (a)...Ch. 2.2 - Use the given graph of f to state the value of...Ch. 2.2 - For the function f whose graph is given, state the...Ch. 2.2 - For the function h whose graph is given, state the...Ch. 2.2 - For the function g whose graph is given, state the...Ch. 2.2 - For the function A whose graph is shown, state the...Ch. 2.2 - For the function f whose graph is shown, state the...Ch. 2.2 - Prob. 10E Ch. 2.2 - Sketch the graph of the function and use it to...Ch. 2.2 - Sketch the graph of the function and use it to...Ch. 2.2 - Prob. 13E Ch. 2.2 - Prob. 14E Ch. 2.2 - Prob. 15E Ch. 2.2 - Sketch the graph of an example of a function f...Ch. 2.2 - Sketch the graph of an example of a function f...Ch. 2.2 - Prob. 18E Ch. 2.2 - Prob. 19E Ch. 2.2 - Guess the value of the limit (if it exists) by...Ch. 2.2 - Prob. 21E Ch. 2.2 - Prob. 22E Ch. 2.2 - Prob. 23E Ch. 2.2 - Prob. 24E Ch. 2.2 - Prob. 25E Ch. 2.2 - Prob. 26E Ch. 2.2 - Prob. 27E Ch. 2.2 - Prob. 28E Ch. 2.2 - Prob. 29E Ch. 2.2 - Prob. 30E Ch. 2.2 - Prob. 31E Ch. 2.2 - Prob. 32E Ch. 2.2 - Determine the infinite limit. limx12x(x1)2 Ch. 2.2 - Prob. 34E Ch. 2.2 - Prob. 35E Ch. 2.2 - Prob. 36E Ch. 2.2 - Prob. 37E Ch. 2.2 - Prob. 38E Ch. 2.2 - Prob. 39E Ch. 2.2 - Prob. 40E Ch. 2.2 - Prob. 41E Ch. 2.2 - Prob. 42E Ch. 2.2 - Prob. 43E Ch. 2.2 - Prob. 44E Ch. 2.2 - Determine limx11x31 and limx1+1x31 (a) by...Ch. 2.2 - Prob. 46E Ch. 2.2 - (a) Estimate the value of the limit limx0 (1 +...Ch. 2.2 - Prob. 49E Ch. 2.2 - Prob. 50E Ch. 2.2 - Prob. 51E Ch. 2.2 - Prob. 52E Ch. 2.2 - Prob. 53E Ch. 2.2 - Prob. 54E Ch. 2.2 - Prob. 55E Ch. 2.3 - Given that limx2f(x)=4limx2g(x)=2limx2h(x)=0 find...Ch. 2.3 - Tire graphs of f and g are given. Use them to...Ch. 2.3 - Prob. 3E Ch. 2.3 - Prob. 4E Ch. 2.3 - Prob. 5E Ch. 2.3 - Prob. 6E Ch. 2.3 - Prob. 7E Ch. 2.3 - Prob. 8E Ch. 2.3 - Prob. 9E Ch. 2.3 - (a) What is wrong with the following equation?...Ch. 2.3 - Prob. 11E Ch. 2.3 - Evaluate the limit, if it exists. limx3x2+3xx2x12 Ch. 2.3 - Prob. 13E Ch. 2.3 - Prob. 14E Ch. 2.3 - Prob. 15E Ch. 2.3 - Evaluate the limit, if it exists....Ch. 2.3 - Prob. 17E Ch. 2.3 - Prob. 18E Ch. 2.3 - Prob. 19E Ch. 2.3 - Prob. 20E Ch. 2.3 - Prob. 21E Ch. 2.3 - Prob. 22E Ch. 2.3 - Prob. 23E Ch. 2.3 - Evaluate the limit, if it exists. limh0(3+h)131h Ch. 2.3 - Prob. 25E Ch. 2.3 - Prob. 26E Ch. 2.3 - Prob. 27E Ch. 2.3 - Prob. 28E Ch. 2.3 - Prob. 29E Ch. 2.3 - Evaluate the limit, if it exists. limx4x2+95x+4 Ch. 2.3 - Prob. 31E Ch. 2.3 - Evaluate the limit, if it exists. limh01(xh)21x2h Ch. 2.3 - Prob. 33E Ch. 2.3 - Prob. 34E Ch. 2.3 - Prob. 35E Ch. 2.3 - Prob. 36E Ch. 2.3 - If 4x 9 f(x) x2 4x + 7 for x 0, find limx4f(x)Ch. 2.3 - If 2x g(x) x4 x2 + 2 for all x, evaluate...Ch. 2.3 - Prove that limx0x4cos2x=0.Ch. 2.3 - Prob. 40E Ch. 2.3 - Prob. 41E Ch. 2.3 - Find the limit, if it exists. 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For the function g whose graph is given, state the...Ch. 2.6 - Prob. 5E Ch. 2.6 - Prob. 6E Ch. 2.6 - Prob. 7E Ch. 2.6 - Prob. 8E Ch. 2.6 - Prob. 9E Ch. 2.6 - Sketch the graph of an example of a function f...Ch. 2.6 - Prob. 11E Ch. 2.6 - Prob. 12E Ch. 2.6 - Prob. 13E Ch. 2.6 - Prob. 14E Ch. 2.6 - Prob. 15E Ch. 2.6 - Prob. 16E Ch. 2.6 - Prob. 17E Ch. 2.6 - Prob. 18E Ch. 2.6 - Prob. 19E Ch. 2.6 - Prob. 20E Ch. 2.6 - Prob. 21E Ch. 2.6 - Prob. 22E Ch. 2.6 - Prob. 23E Ch. 2.6 - Prob. 24E Ch. 2.6 - Prob. 25E Ch. 2.6 - Prob. 26E Ch. 2.6 - Prob. 27E Ch. 2.6 - Prob. 28E Ch. 2.6 - Prob. 29E Ch. 2.6 - Prob. 30E Ch. 2.6 - Prob. 31E Ch. 2.6 - Prob. 32E Ch. 2.6 - Prob. 33E Ch. 2.6 - Prob. 34E Ch. 2.6 - Prob. 35E Ch. 2.6 - Prob. 36E Ch. 2.6 - Prob. 37E Ch. 2.6 - Prob. 38E Ch. 2.6 - Prob. 39E Ch. 2.6 - Prob. 40E Ch. 2.6 - Prob. 41E Ch. 2.6 - Prob. 42E Ch. 2.6 - Prob. 43E Ch. 2.6 - Prob. 44E Ch. 2.6 - (a) Estimate the value of limx(x2+x+1+x) by...Ch. 2.6 - Prob. 46E Ch. 2.6 - Prob. 47E Ch. 2.6 - Find the horizontal and vertical asymptotes of...Ch. 2.6 - Prob. 49E Ch. 2.6 - Prob. 50E Ch. 2.6 - Prob. 51E Ch. 2.6 - Prob. 52E Ch. 2.6 - Prob. 53E Ch. 2.6 - Prob. 54E Ch. 2.6 - Prob. 55E Ch. 2.6 - Prob. 56E Ch. 2.6 - Find a formula for a function f that satisfies the...Ch. 2.6 - Prob. 58E Ch. 2.6 - A function f is a ratio of quadratic functions and...Ch. 2.6 - Prob. 60E Ch. 2.6 - Prob. 61E Ch. 2.6 - Prob. 62E Ch. 2.6 - Prob. 63E Ch. 2.6 - Prob. 64E Ch. 2.6 - Prob. 65E Ch. 2.6 - Prob. 66E Ch. 2.6 - Prob. 67E Ch. 2.6 - Prob. 68E Ch. 2.6 - Prob. 69E Ch. 2.6 - Prob. 70E Ch. 2.6 - Prob. 71E Ch. 2.6 - Prob. 72E Ch. 2.6 - Prob. 73E Ch. 2.6 - Prob. 74E Ch. 2.6 - Prob. 75E Ch. 2.6 - Prob. 76E Ch. 2.6 - Prob. 77E Ch. 2.6 - Prob. 78E Ch. 2.6 - Prob. 79E Ch. 2.6 - Prob. 80E Ch. 2.6 - Prob. 81E Ch. 2.7 - A curve has equation y = f(x) (a) Write an...Ch. 2.7 - Graph the curve y = ex in the viewing rectangles [...Ch. 2.7 - Prob. 3E Ch. 2.7 - Prob. 4E Ch. 2.7 - Find an equation of the tangent line to the curve...Ch. 2.7 - Prob. 6E Ch. 2.7 - Prob. 7E Ch. 2.7 - 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Prob. 43E Ch. 2.7 - Prob. 44E Ch. 2.7 - Prob. 45E Ch. 2.7 - Prob. 46E Ch. 2.7 - Prob. 47E Ch. 2.7 - Prob. 48E Ch. 2.7 - Prob. 49E Ch. 2.7 - The table shows values of the viral load V(r) in...Ch. 2.7 - Prob. 51E Ch. 2.7 - Prob. 52E Ch. 2.7 - Prob. 53E Ch. 2.7 - Prob. 54E Ch. 2.7 - Prob. 55E Ch. 2.7 - Prob. 56E Ch. 2.7 - The quantity of oxygen that can dissolve in water...Ch. 2.7 - The graph shows the influence of the temperature T...Ch. 2.7 - Prob. 59E Ch. 2.7 - Prob. 60E Ch. 2.7 - (a) Graph the function f(x)=sinx11000sin(1000x) in...Ch. 2.8 - Use the given graph to estimate the value of each...Ch. 2.8 - Prob. 2E Ch. 2.8 - Match the graph of each function in (a)(d) with...Ch. 2.8 - Trace or copy the graph of the given function .f....Ch. 2.8 - Trace or copy the graph of the given function .f....Ch. 2.8 - Prob. 6E Ch. 2.8 - Trace or copy the graph of the given function .f....Ch. 2.8 - Trace or copy the graph of the given function .f....Ch. 2.8 - Trace or copy the graph of the given function .f....Ch. 2.8 - Trace or copy the graph of the given function .f....Ch. 2.8 - Prob. 11E Ch. 2.8 - Prob. 12E Ch. 2.8 - Prob. 13E Ch. 2.8 - Prob. 14E Ch. 2.8 - The graph shows how the average age of first...Ch. 2.8 - Prob. 16E Ch. 2.8 - Prob. 17E Ch. 2.8 - Prob. 18E Ch. 2.8 - Prob. 19E Ch. 2.8 - Prob. 20E Ch. 2.8 - Prob. 21E Ch. 2.8 - Prob. 22E Ch. 2.8 - Prob. 23E Ch. 2.8 - Prob. 24E Ch. 2.8 - Prob. 25E Ch. 2.8 - Prob. 26E Ch. 2.8 - Prob. 27E Ch. 2.8 - Prob. 28E Ch. 2.8 - Prob. 29E Ch. 2.8 - Prob. 30E Ch. 2.8 - Prob. 31E Ch. 2.8 - Prob. 32E Ch. 2.8 - Prob. 33E Ch. 2.8 - Prob. 34E Ch. 2.8 - Prob. 35E Ch. 2.8 - Prob. 36E Ch. 2.8 - Prob. 37E Ch. 2.8 - Water temperature affects the growth rate of brook...Ch. 2.8 - Let P represent the percentage of a city's...Ch. 2.8 - Prob. 40E Ch. 2.8 - Prob. 41E Ch. 2.8 - Prob. 42E Ch. 2.8 - Prob. 43E Ch. 2.8 - Prob. 44E Ch. 2.8 - Prob. 45E Ch. 2.8 - Prob. 46E Ch. 2.8 - Prob. 47E Ch. 2.8 - Prob. 48E Ch. 2.8 - Prob. 49E Ch. 2.8 - Prob. 50E Ch. 2.8 - Prob. 51E Ch. 2.8 - Prob. 52E Ch. 2.8 - 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If f ( x ) = 〚 x 〛 + 〚 − x 〛 , then lim x → 2 f ( x ) exists but is not equal to f ( 2 ) .

Solution Summary: The author explains that x= the largest integer that is less than or equal to x.

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To show: If f(x)=〚x〛+〚−x〛 , then limx→2f(x) exists but is not equal to f(2).

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Chapter 2 Solutions