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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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 limxaf(x)=f(a).

Theorem 1:

The limit limxaf(x)=L if and only if limxaf(x)=L=limxa+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 2x<1.

limx2+x=limx2+(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 limxax exist.

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