   Chapter 2.3, Problem 55E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# If f ( x ) = 〚 x 〛 + 〚 − x 〛 , show that lim x→0 f(x) exists hut is not equal to f(2).

To determine

To show: If f(x)=x+x , then limx2f(x) exists but is not equal to f(2).

Explanation

Definition used:

Greatest integer function:x= the largest integer that is less than or equal to x.

Theorem 1:

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

limx2f(x)=limx2(x+x)

Obtain the limit of the function f(x)=x+x as x approaches left-hand side of 2.

limx2f(x)=limx2(x+x)=limx2x+limx2x[by limit law 1]

From the Figure 1 and 2, it is observed that limx2x=1 and limx2x=2.

limx2x+limx2x=1+(2)=12=1

Obtain the limit of the function f(x)=x+x as x approaches right-hand side of 2.

limx2+f(x)=limx2+(x+x)=limx2+x+limx2+x[by limit law 1]

From the Figure 1 and 2, it is observed that limx2+x=2 and limx2+x=3.

limx2+x+limx2+x=2+(3)=23=1

Therefore, the left and right side limits are equal

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 