Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2}\left(x^{2}-3 x+1\right) $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The limit is the value that the function appears to be approaching as \(x\) approaches 2, from both the left-hand side and the right-hand side, according to the table of values and the graph of the function.

Step 1: Complete the Table

First, you'll need to use the function \(f(x) = x^{2}-3 x+1\) to calculate the values for the table. For each \(x\) value: 1.9, 1.99, 1.999, 2.001, 2.01, 2.1, you have to substitute the \(x\) value into function and find the corresponding \(f(x)\).

Step 2: Identify the Trend

After completing the table, look for a trend in the both the left-hand limit and the right-hand limit of the function as \(x\) approaches 2. This will help to estimate the limit.

Step 3: Graph the Function

You must also graph the function \(f(x) = x^{2}-3 x+1\) using a graphing utility, to visually confirm your result.

Step 4: Estimate the Limit

Based on the trends in the table and the shape of the graph, you must then estimate the limit of the function as \(x\) approaches 2.

Key Concepts

Graphing Functions

Graphing functions is a fundamental tool for visualizing mathematical behavior. It allows us to see how a function behaves across an interval by plotting all potential outputs given a specific input range. In the given exercise, once you have calculated the table of values for the function \(f(x) = x^{2}-3 x+1\), using a graphing utility to plot these points can greatly aid in understanding. Graphs provide a clear, visual way to predict limits and identify trends.
Using a graph:

• Makes it easier to see where the function is approaching a specific value.
• Helps identify any discontinuities or behaviors that could affect limit estimation.
• Offers a visual confirmation of numerical calculations, adding a layer of understanding.

In our case, looking at the graph of \(f(x) = x^{2}-3 x+1\) as \(x\) approaches 2 will confirm the convergence of values that the numerical table suggests. Visualizing the function alongside numerical data solidifies your grasp on how graphing aids in understanding and estimating limits.

Limit Estimation

Limit estimation requires looking at the behavior of a function as it approaches a particular value, in this case, 2. Completing the table for different values of \(x\) near 2, helps observe how \(f(x)\) behaves.
When estimating limits:

• Look for patterns or trends in values as \(x\) approaches the specified number from the left and right.
• Consider both sides because a proper limit requires the function to approach the same value from both directions.
• Keep a close eye on the numeric trend to see if the values stabilize, indicating the limit.

For \(f(x) = x^{2}-3 x+1\), as we fill in the table, observing whether the values of \(f(x)\) near \(x=2\) converge helps us to predict the behavior precisely at that point. When these numbers start to stabilize as they approach a point, the limit is likely this value. This method supports you in making educated predictions about limits.

Polynomial Functions

Polynomial functions like \(f(x) = x^{2}-3 x+1\) are expressions consisting of variables and coefficients involving only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. In terms of limits, polynomial functions are generally straightforward, as they tend to be continuous across their domain.
Understanding the properties of polynomials:

• They are continuous and smooth, which simplifies the identification of limits as there are no breaks or jumps.
• The limit at a point where the function is defined typically equals the function's evaluation at that point.
• Polynomials follow predictable patterns, easing the analysis of their behavior around specific values.

In our exercise, evaluating the limit of \(f(x) = x^{2}-3 x+1\) as \(x\) approaches 2 involves calculating \(f(2)\). For polynomial functions, this direct substitution method is not only effective but confirms what the table suggests regarding limit behavior. This characteristic makes polynomial functions a great starting point for studying calculus concepts.