Problem 38 - Precalculus

Question

See if a table of values suggests a limit exists for the functions and approaches indicated. \(g(x)=6 x^{2}-x\) as \(x \rightarrow-\frac{1}{2}\) from the right.

Step-by-Step Solution

Answer

The limit of \( g(x) \) as \( x \rightarrow -\frac{1}{2}^+ \) is 2.

Step 1: Understanding the Function and Direction of Approaching

To determine if a limit exists, we need to inspect the behavior of the function \( g(x) = 6x^2 - x \) as \( x \) approaches \( -\frac{1}{2} \) from the right. This means that we need to determine the values of \( g(x) \) for values of \( x \) slightly greater than \( -\frac{1}{2} \).

Step 2: Creating the Table of Values

Choose a series of \( x \) values slightly larger than \( -\frac{1}{2} \) like \( -0.4, -0.45, -0.49, -0.495 \). Then, calculate \( g(x) \) for each of these points:\[\begin{align*}g(-0.4) & = 6(-0.4)^2 - (-0.4) = 0.96 + 0.4 = 1.36, \g(-0.45) & = 6(-0.45)^2 - (-0.45) = 1.215 + 0.45 = 1.665, \g(-0.49) & = 6(-0.49)^2 - (-0.49) = 1.4414 + 0.49 = 1.9314, \g(-0.495) & = 6(-0.495)^2 - (-0.495) = 1.47015 + 0.495 = 1.96515.\end{align*}\]

Step 3: Observing the Values

From the table of values, observe the trend as \( x \) approaches \(-\frac{1}{2} \) from the right. For \( x = -0.4, -0.45, -0.49, \) and \(-0.495\), the corresponding \( g(x) \) values are \( 1.36, 1.665, 1.9314, \) and \( 1.96515 \) respectively.

Step 4: Determining the Limit

Noting the trend of the values, as \( x \) gets closer to \(-\frac{1}{2}\) from the right, \( g(x) \) approaches 2. Thus, we can infer that the limit of \( g(x) \) as \( x \) approaches \(-\frac{1}{2}\) from the right is 2.

Key Concepts

Approaching Values

When we talk about the concept of approaching values in calculus, we refer to understanding what happens to the output of a function as the input gets extremely close to a specific point, but doesn’t necessarily reach it. This is essentially what limits are all about. If we have the function \( g(x) = 6x^2 - x \), and we want to find out what happens to this function as \( x \) approaches \(-\frac{1}{2}\) from the right, it means we are interested in the behavior of \( g(x) \) for values of \( x \) that are slightly larger than \(-\frac{1}{2}\).

• Think of approaching values like a car approaching a stop. You can predict when it will come to a stop based on its speed and distance from the stop sign even before it actually stops.
• Approaching values are essential in determining the limit of a function, which helps us in understanding the behavior of the function at points it may not explicitly be defined.

Calculating how the values change as \( x \) gets closer to \(-\frac{1}{2}\) gives us insight into the trend of the function and whether a finite limit exists.

Table of Values

In mathematics, and particularly in calculus, a table of values is a handy tool for visualizing the behavior of a function. To comprehend the trend of \( g(x) = 6x^2 - x \) as \( x \) is approaching \(-\frac{1}{2}\) from the right, one effective method is to calculate a set of such values, forming a table.

• By choosing \( x \) values like \(-0.4, -0.45, -0.49, \) and \(-0.495\), we can compute the corresponding \( g(x) \) values to observe any pattern as \( x \) increases towards \(-\frac{1}{2}\).
• This direct computation is crucial as it allows us to see explicitly what happens to the value of \( g(x) \) as \( x \) trends towards a particular point.

Using a table helps make the abstract concept of limits more concrete by drawing attention to numerical evidence for a limit existing, rather than speculative conclusions.

Precalculus

Precalculus serves as a foundation for calculus, introducing concepts essential for understanding and deriving limits. Precalculus involves topics like functions, graphs, and understanding how values behave in relation to these functions. In this exercise on limits of a function, we are applying precalculus concepts to carefully select approaching points \( x \) and calculate \( g(x) \).

• This involves understanding the function’s formula and how changes in \( x \) impact \( g(x) \).
• Knowledge from precalculus aids us in evaluating the limit by focusing on how function values evolve, which signifies how crucial precalculus knowledge is to understand calculus concepts.

Therefore, a strong grasp of precalculus principles is indispensable for properly evaluating limits, preparing you to dive deeper into more complex calculus topics.