BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2.3, Problem 8E

(a)

To determine

To explain: The equation x2+x6x2=x+3 is incorrect.

Expert Solution

Explanation of Solution

Let the function f(x)=x2+x6x2 and g(x)=x+3.

Construct the table of f(x) and g(x) for some arbitrary values of x.

xf(x)g(x)
−3f(3)=(3)2+(3)6(3)2=0g(3)=3+3=0
−2f(2)=(2)2+(2)6(2)2=1g(2)=2+3=1
−1f(1)=(1)2+(1)6(1)2=2g(1)=1+3=2
0f(0)=(0)2+(0)6(0)2=3g(0)=0+3=3
1f(1)=(1)2+(1)6(1)2=4g(1)=1+3=4
2f(2)=(2)2+(2)6(2)2=00g(2)=2+3=5
3f(3)=(3)2+(3)6(3)2=6g(3)=3+3=6

From the table, it is observed that, f(x) is not defined at x=2 but g(x)  is defined at x=2.

The domain of f(x)={x2+x6x2:x except x=2} and the domain of g(x)={x+3:x}.

In general, the equation holds for all x not equal to 2. That is, f(x)=g(x) for x except x=2.

Therefore, f(x) and g(x) are not equal.

That is, x2+x6x2x+3.

(b)

To determine

To explain: The equation limx2x2+x6x2=limx2(x+3) is correct.

Expert Solution

Explanation of Solution

From part (a), the equation holds for all real numbers except 2.

That is, x2+x6x2=x+3 is true when x2.

limx2x2+x6x2=limx2(x2)(x+3)x2=limx2(x+3)

Thus, it can be concluded that both the limit functions are equal.

That is, limx2x2+x6x2=limx2(x+3).

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