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Math
CalculusSingle Variable Calculus: Early Transcendentals8th Edition

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Chapter 2.3, Problem 53E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

(a) If the symbol 〚 〛 denotes the greatest integer function defined in Example I 0, evaluate

(i) lim x → 2 + 〚 x 〛

(ii) lim x → − 2 〚 x 〛

(iii) lim x → 2.4 〚 x 〛

(b) If n is an integer, evaluate

(i) lim x → n − 〚 x 〛

(ii) lim x → n + 〚 x 〛

(c) For what values of a does lim x → 0 〚 x 〛 exist?

(a)

To determine

To evaluate: The limit of the greatest integer function.

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 limx→af(x)=f(a).

Theorem 1:

The limit limx→af(x)=L if and only if limx→a−f(x)=L=limx→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.

limx→−2+〚x〛=limx→−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 limx→a〚x〛 exist.

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Sect-2.1 P-1ESect-2.1 P-2ESect-2.1 P-3ESect-2.1 P-4ESect-2.1 P-5ESect-2.1 P-6ESect-2.1 P-7ESect-2.1 P-8ESect-2.1 P-9ESect-2.2 P-1E
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