(a) What is wrong with the following equation? x^2 + x - 6/x - 2 = x + 3 (b) In view of part...


(a) What is wrong with the following equation? {eq}x^2 + x - \frac6x - 2 = x +3 {/eq}

(b) In view of part (a), explain why the equation {eq}\lim_{x \to 2} (x^2 + x - \frac 6x - 2) = \lim_{x \to 2} (x+3) {/eq} is correct.

Equivalent Functions:

{eq}\\ {/eq}

Suppose we have two equivalent functions i.e. {eq}f(x)=g(x) {/eq} but {eq}f(x) {/eq} is undefined for some {eq}x {/eq} and at the same time {eq}g(x) {/eq} is defined for that {eq}x {/eq}, so we have {eq}f(x)\neq g(x) {/eq} at that value of {eq}x. {/eq}

Answer and Explanation:

{eq}\\ {/eq}

(a) Let {eq}f(x)=x^2 + x - \dfrac {6} {x} - 2 {/eq} and {eq}g(x)=x+3 {/eq}.

We can clearly see that {eq}f(x) {/eq} is undefined for {eq}x=0 {/eq} but {eq}g(x)=3 {/eq} at {eq}x=0 {/eq}, which is defined.

Hence, at {eq}x=0, \ f(x)\neq g(x) {/eq}.

(b) We have : {eq}\lim_{x \to 2} (x^2 + x - \dfrac {6}{x} - 2)=2^2+2-3-2=6-5=1 \ \& \ \lim_{x \to 2} (x+3)=2+3=5 {/eq}, so they are not equal.

Learn more about this topic:

Expressions of Rational Functions

from Precalculus: High School

Chapter 13 / Lesson 4

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