Problem 37 - Precalculus

Question

See if a table of values suggests a limit exists for the functions and approaches indicated. \(f(x)=2 x^{2}-3 x\) as \(x \rightarrow-3\) from the right.

Step-by-Step Solution

Answer

As \( x \to -3^+ \), \( f(x) \to 18 \).

Step 1: Evaluate the Function

First, we consider the function \( f(x) = 2x^2 - 3x \). To investigate the limit as \( x \to -3 \) from the right, substitute several values slightly greater than \(-3\) into the function to predict the behavior.

Step 2: Choice of x-values

Choose values for \( x \) just to the right of \(-3\). For example, we could use \( x = -2.9, -2.95, -2.99 \). Compute \( f(x) \) for each of these values to see how the function behaves.

Step 3: Compute Function Values

Calculate \( f(-2.9) = 2(-2.9)^2 - 3(-2.9) = 16.91 \), \( f(-2.95) = 2(-2.95)^2 - 3(-2.95) = 16.9025 \), and \( f(-2.99) = 2(-2.99)^2 - 3(-2.99) = 16.9801 \). Observe how these values change as \( x \) approaches \(-3\) from the right.

Step 4: Analyze Behavior of Function

Look at the results: as \( x \to -3 \), the function values \( 16.91, 16.9025, 16.9801 \) appear to approach a certain value. This suggests that as \( x \to -3 \) from the right, \( f(x) \to 18 \).

Step 5: Conclusion on Limit

Based on the calculated values, it appears that the limit of \( f(x) \) as \( x \) approaches \(-3\) from the right is indeed \( 18 \).

Key Concepts

Right-Hand Limit

A right-hand limit evaluates the behavior of a function as it approaches a specific point from the right. In this case, we're interested in what happens to the function \( f(x) = 2x^2 - 3x \) as \( x \) approaches \(-3\) from values slightly greater than \(-3\). When computing right-hand limits, we look for what value the function is tending towards when coming from the right.
In the context of this exercise, it involves choosing x-values that are incrementally closer to \(-3\) but still greater, such as \(-2.9, -2.95,\) and \(-2.99\). The process helps to observe if these values collectively suggest a particular limit value that \( f(x) \) seems to settle towards.

Function Evaluation

Function evaluation is a process where you substitute particular x-values into the function to compute the corresponding y-values or outputs. For \( f(x) = 2x^2 - 3x \), we substitute chosen values like \(-2.9, -2.95,\) and \(-2.99\) into the function to see what outputs are generated.
This step is critical because it allows us to concretely see how \( f(x) \) behaves as \( x \) approaches \(-3\) from the right. By calculating \( f(-2.9) = 16.91 \), \( f(-2.95) = 16.9025 \), and \( f(-2.99) = 16.9801 \), one can see a trend in the values. This consistent decrease in function outputs implies a limit that the function approaches, helping us ultimately assess the right-hand limit.

Table of Values Analysis

Analyzing a table of values involves examining the computed function outputs as they relate to the input values close to the limit point. In our example, after evaluating \( f(x) \) for \( x = -2.9, -2.95,\) and \(-2.99\), we obtained outputs \( 16.91, 16.9025, \) and \( 16.9801 \).

• Notice the gradual approach to a specific value as x approaches \(-3\).
• The aim is to see if \( f(x) \) approaches a particular number, in this case, suggesting that \( f(x) \) gets closer to \( 18 \).

Using a table of values allows for a visual check to support mathematical reasoning, thereby making limit assessment clearer. It helps provide confirmation of our calculated expectation that \( f(x) \to 18 \) as \( x \to -3 \) from the right.