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Chapter 11: Problem 97

What value does \(R(x)=\frac{3 x^{2}+14 x+8}{x^{2}+x-12}\) approach as \(x \rightarrow-4 ?\)

Short Answer

Expert verified
The value that \( R(x) \) approaches as \( x \rightarrow -4 \) is \( \frac{10}{7} \).

Step by step solution

01

Factor the Numerator

Factor the numerator of the rational function. The numerator is given by \( 3x^2 + 14x + 8 \). By factoring, we get: \( 3x^2 + 14x + 8 = (3x + 2)(x + 4) \).
02

Factor the Denominator

Next, factor the denominator of the rational function. The denominator is given by \( x^2 + x - 12 \). By factoring, we get: \( x^2 + x - 12 = (x + 4)(x - 3) \).
03

Simplify the Rational Function

Now substitute the factored forms into the original rational function: \( R(x) = \frac{(3x + 2)(x + 4)}{(x + 4)(x - 3)}\). Notice that \( x + 4 \) is a common factor in the numerator and the denominator, so it cancels out: \( R(x) = \frac{3x + 2}{x - 3} \)
04

Evaluate the Limit as x Approaches -4

To find the value that \( R(x) \) approaches as \( x \) approaches \( -4 \), substitute \( x = -4 \) into the simplified rational function: \( R(x) = \frac{3(-4) + 2}{-4 - 3} = \frac{-12 + 2}{-7} = \frac{-10}{-7} = \frac{10}{7} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Polynomials
Factoring polynomials is a key skill in algebra. It involves breaking down a polynomial into simpler pieces called factors. For the given exercise, we started by factoring the numerator, which is a polynomial \(3x^2 + 14x + 8\). We look for two numbers that multiply to give the constant term (8) and add to give the middle term (14). We found that the factors are \((3x + 2)(x + 4)\).

Similarly, we factor the denominator \(x^2 + x - 12\). We look for two numbers that multiply to give -12 and add to give 1. This gives us \((x + 4)(x - 3)\). Factoring helps in breaking down complex expressions into simpler, manageable parts.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing them to their simplest form. For our function \(R(x) = \frac{3x^2 + 14x + 8}{x^2 + x - 12}\), we first factor the numerator and the denominator.

The factored form is \(\frac{(3x + 2)(x + 4)}{(x + 4)(x - 3)}\). Notice that \(x + 4\) is a common factor in both the numerator and the denominator. We can cancel these out as long as \(x eq -4\), to avoid division by zero.

After canceling, we have \(R(x) = \frac{3x + 2}{x - 3}\). This simplified form is easier to work with for further analysis, such as evaluating limits.
Evaluating Limits
Evaluating limits helps us find the value that a function approaches as the input approaches a certain point. For our exercise, we needed to find the limit of \(R(x) = \frac{3x + 2}{x - 3}\) as \(x\) approaches \(-4\).

In this process, we simply substituted \(-4\) for \(x\) in the simplified rational function. Substituting \( x = -4 \) into \( \frac{3x + 2}{x - 3} \) we get:

\[ R(x) = \frac{3(-4) + 2}{-4 - 3} = \frac{-12 + 2}{-7} = \frac{-10}{-7}. \]
Since the result is \( \frac{10}{7} \), the value \(R(x)\) approaches as \(x\) approaches \(-4\) is \( \frac{10}{7} \). Evaluating limits is crucial for understanding the behavior of functions near specified points.

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Most popular questions from this chapter

Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time the first person, standing at point \(A,\) hears the burst. One second later the second person, standing at point \(B,\) hears the burst. If the person at point \(B\) is due west of the person at point \(A\), and if the display is known to occur due north of the person at point \(A,\) where did the fireworks display occur?

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