Question

(a) what is wrong with the following equation?
\\(\frac{x^{2}+x - 42}{x - 6}=x + 7\\)
\\( (x - 6)(x + 7)
eq x^{2}+x - 42\\)
the left - hand side is not defined for (x = 0), but the right - hand side is.
the left - hand side is not defined for (x = 6), but the right - hand side is.
none of these — the equation is correct.

(b) in view of part (a), explain why the equation
\\(\lim_{x\to6}\frac{x^{2}+x - 42}{x - 6}=\lim_{x\to6}(x + 7)\\)
is correct.
since \\(\frac{x^{2}+x - 42}{x - 6}\\) and (x + 7) are both continuous, the equation follows.
since the equation holds for all (x
eq6), it follows that both sides of the equation approach the same limit as (x\to6).
this equation follows from the fact that the equation in part (a) is correct.
none of these — the equation is not correct.

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Explanation:

Step1: Analyze domain of left - hand side

For the rational function $\frac{x^{2}+x - 42}{x - 6}$, the denominator cannot be zero. Setting $x-6 = 0$, we get $x = 6$. So the left - hand side is undefined at $x = 6$. The right - hand side $y=x + 7$ is a polynomial and is defined for all real $x$.

Step2: Analyze limit equality

The function $\frac{x^{2}+x - 42}{x - 6}=\frac{(x - 6)(x+7)}{x - 6}=x + 7$ for $x
eq6$. When taking the limit as $x
ightarrow6$, since the two functions $y=\frac{x^{2}+x - 42}{x - 6}$ and $y=x + 7$ are equal for all $x
eq6$, the limits of both functions as $x
ightarrow6$ are the same.

Answer:

(a) The left - hand side is not defined for $x = 6$, but the right - hand side is.
(b) Since the equation holds for all $x
eq6$, it follows that both sides of the equation approach the same limit as $x
ightarrow6$.