(a) Explanation Given: Definition used: Greatest integer function: The largest integer that is less than or equal to x is, 〚 x 〛 Direct substitution property: If f is a polynomial or a rational function and a is in the domain of f , then lim x → a f ( x ) = f ( a ) . 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 ) . Calculation: Section (i) Obtain the limit of the greatest integer function as x approaches right-hand side of −2. Define the greatest integer function as x approaches right-hand side of −2. 〚 x 〛 = − 2 for − 2 ≤ x < − 1 . lim x → − 2 + 〚 x 〛 = lim x → − 2 + ( − 2 ) = − 2 [ by direct substitution ] Thus, the limit of the greatest integer function as x approaches right-hand side of −2 is −2. Section (ii) Obtain the limit of the greatest integer function as x approaches −2. From section (i), the limit of the greatest integer function as x approaches right-hand side of −2 is −2. Obtain the limit of the greatest integer function as x approaches left-hand side of −2 is −3. Define the greatest integer function as x approaches left-hand side of −2 (b) To determine To evaluate: The limit of the greatest integer function. (c) To determine To find: The values of a does lim x → a 〚 x 〛 exist.

2nd Edition

ISBN: 9781133490975

Author: Stewart, James

Publisher: Cengage Learning

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Limits

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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Chapter 1.4, Problem 45E

(a)

To determine

To evaluate: The limit of the greatest integer function.

(b)

To determine

To evaluate: The limit of the greatest integer function.

(c)

To determine

To find: The values of a does limx→a〚x〛 exist.

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Chapter 1.4, Problem 44E

Chapter 1.4, Problem 46E

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Chapter 1 Solutions

Essential Calculus

Ch. 1.1 - 1. If f(x)=x+2x and g(u)=u+2u, is it true that f =...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 - Prob. 5E 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. 9E Ch. 1.1 - The graph shows the height of the water in a...

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Prob. 38E Ch. 1.6 - Prob. 36E Ch. 1.6 - Find the horizontal and vertical asymptotes of...Ch. 1.6 - Prob. 39E Ch. 1.6 - Prob. 34E Ch. 1.6 - Let P and Q be polynomials. 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The limit of the greatest integer function.

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To evaluate: The limit of the greatest integer function.

To evaluate: The limit of the greatest integer function.

To find: The values of a does limx→a〚x〛 exist.

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Chapter 1 Solutions