Problem 32 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$v(x)=\frac{x^{2}-3 x-10}{2 x+4} ; x \rightarrow-2$$

Step-by-Step Solution

Answer

As \(x\) approaches \(-2\), \(v(x)\) approaches \(-0.5\) with limit notation: \(\lim_{{x \to -2}} \frac{x^{2} - 3x - 10}{2x + 4} = -0.5\).

Step 1: Create a Table of Values

Choose values of \(x\) that are around \(-2\), such as \(-2.1, -2.01, -2.001, -1.9, -1.99, -1.999\), and substitute them into the function \(v(x)=\frac{x^{2}-3x-10}{2x+4}\). Calculate \(v(x)\) for each of these values to help understand how \(v(x)\) behaves as \(x\) approaches \(-2\).

Step 2: Calculate the Function Values

Use the chosen \(x\) values to calculate \(v(x)\):\[ v(-2.1) = \frac{(-2.1)^{2} - 3(-2.1) - 10}{2(-2.1) + 4} \approx -0.525 \]\[ v(-2.01) = \frac{(-2.01)^{2} - 3(-2.01) - 10}{2(-2.01) + 4} \approx -0.5025 \]\[ v(-2.001) = \frac{(-2.001)^{2} - 3(-2.001) - 10}{2(-2.001) + 4} \approx -0.50025 \]\[ v(-1.9) = \frac{(-1.9)^{2} - 3(-1.9) - 10}{2(-1.9) + 4} \approx -0.475 \]\[ v(-1.99) = \frac{(-1.99)^{2} - 3(-1.99) - 10}{2(-1.99) + 4} \approx -0.4975 \]\[ v(-1.999) = \frac{(-1.999)^{2} - 3(-1.999) - 10}{2(-1.999) + 4} \approx -0.49975 \]

Step 3: Analyze the Values

Observe the pattern in the calculated \(v(x)\) values as \(x\) gets closer to \(-2\). The values \(-0.525, -0.5025, -0.50025, -0.475, -0.4975, -0.49975\) approach \(-0.5\). This suggests that as \(x\) approaches \(-2\), \(v(x)\) approaches \(-0.5\).

Step 4: Write the Limit Notation and Explanation

Based on the analysis in Step 3, the limit of \(v(x)\) as \(x\) approaches \(-2\) is \(-0.5\). In limit notation, this is written as:\[ \lim_{{x \to -2}} \frac{x^{2} - 3x - 10}{2x + 4} = -0.5 \]In words, as \(x\) approaches \(-2\), the value of the function \(v(x)\) approaches \(-0.5\).

Key Concepts

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. A typical form of a rational function is given by \( f(x) = \frac{P(x)}{Q(x)} \), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). These functions can exhibit interesting behaviors at certain points, often revealing important aspects of their structure. In our specific exercise, the rational function involved is \( v(x) = \frac{x^2 - 3x - 10}{2x + 4} \).
Understanding the nature of rational functions is crucial, particularly how the degree of the numerator and the denominator affects the behavior of the function. For example, when the degree of the numerator is the same as the degree of the denominator, the function tends to level off to a constant value as \(x\) approaches extremely large positive or negative values.
Key aspects of rational functions include:

• Vertical asymptotes, which occur when the denominator is zero, as long as the numerator is not zero at that point too.
• Horizontal or oblique asymptotes, which describe the behavior of the function as \(x\) becomes very large or very small.
• Intercepts, found where the function crosses the axes.

Recognizing these features assists in predicting the behavior of the function across its domain.

Approaching Values

When analyzing rational functions, the concept of approaching values becomes significant, especially in the context of limits. Approaching values means examining the behavior of a function as the input gets arbitrarily close to a particular value, in this case \( x \to -2 \).
In the exercise, we approach the value \(-2\) using specific values of \(x\) that are just a bit greater and a bit less, such as \(-2.1, -2.01, -2.001, -1.9, -1.99, -1.999\). These values help to understand how the function behaves near that point, providing insight into its trend.
Approaching values utilizes concepts such as:

• Limit notation, \( \lim_{x \to c} f(x) \), where you observe the function's value as \(x\) closes in on \(c\).
• The direction of approach, which refers to whether \(x\) approaches from the left or right, or from below or above.
• The outcome or the result value that \(f(x)\) is converging to, helping to confirm the function's continuity or find points of discontinuity.

This analysis uncovers deeper insights into the function's mechanics near the specified \(x\) value.

Function Behavior Analysis

Function behavior analysis is about understanding how a function acts near particular points or ranges. In simpler terms, it is studying the pattern of the output values based on input changes. This concept is relevant when calculating limits.
To analyze the function behavior around \(x = -2\), we calculate \(v(x)\) using a series of values close to \(-2\). As observed in the exercise, calculated values like \(-0.525, -0.5025, -0.50025, -0.475, -0.4975, -0.49975\) steadily approach \(-0.5\).
This pattern reveals important insights, such as:

• Continuity or discontinuity of the function at a point, understanding if the function jumps or is smooth.
• Limiting values which indicate what number the function values are heading towards as \(x\) nears a specific point.
• Potential asymptotic behavior, pointing to values that the function may approach but never quite reaches.

Analyzing these aspects not only aids in limit calculation but also builds a strong foundation for studying more complex functions and their unique behaviors in calculus.