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Chapter 2: Problem 5

Explain why \(\lim _{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}=\lim _{x \rightarrow 3}(x-4)\).

Short Answer

Expert verified
In this problem, we needed to determine if the limit of the given rational function equals the limit of the linear function as x approaches 3. By factoring the numerator of the rational function, we found a common factor that could be cancelled out with the denominator, which simplified the rational function into the linear function \(x-4\). We then calculated the limit of the linear function as x approaches 3, resulting in -1. Therefore, the limit of the given rational function is equal to the limit of the linear function as x approaches 3.

Step by step solution

01

Factor the numerator of the rational function

We are given the rational function \(\frac{x^2 - 7x + 12}{x - 3}\). We need to factor the quadratic expression in the numerator, which is \(x^2 - 7x + 12\). The two numbers whose product is 12 and sum is -7 are -3 and -4. So the factored form of the numerator is \((x-3)(x-4)\).
02

Simplify the rational function

Now that we have factored the numerator, we can simplify the rational function by cancelling out any common factors between the numerator and the denominator. We have: \(\frac{(x-3)(x-4)}{x-3}\) Since \((x-3)\) is a common factor in both the numerator and the denominator, we can cancel it out: \(\frac{(x-3)(x-4)}{x-3} = x-4\)
03

Calculate the limit of the simplified function as x approaches 3

Now that the rational function has been simplified to the linear function \(x-4\), we can evaluate the limit as x approaches 3: \(\lim_{x \rightarrow 3}(x-4)\) To evaluate the limit, we simply substitute the value x = 3 into the expression: \((3-4)=-1\)
04

Conclusion

Since we have simplified the given rational function to the linear function \(x-4\) and calculated the limit of the linear function as x approaches 3, we can conclude that: \(\lim_{x \rightarrow 3}\frac{x^2-7x+12}{x-3} = \lim_{x\rightarrow 3}(x-4)\)

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