# The equation x 2 + x − 6 x − 2 = x + 3 is incorrect. ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### 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+x−6x−2=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.

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

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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