Question

Use a graphing utility or spreadsheet software program to complete the table. Then use the result to estimate the limit of \(f(x)\) as \(x\) approaches infinity. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ \(f(x)=\frac{x^{2}-1}{0.02 x^{2}}\)

Short answer

The limit as \(x\) approaches infinity for the function \(f(x)=\frac{x^{2}-1}{0.02x^{2}}\) is 50.

Step-by-step solution

Substitute the Values of x in the Function

To populate the table with corresponding \(f(x)\) values, substitute the \(x\) values given in the table (i.e., \(10^{0}, 10^{1}, 10^{2}, 10^{3}, 10^{4}, 10^{5}, 10^{6}\)) into the function \(f(x)=\frac{x^{2}-1}{0.02x^{2}}\). For example, when \(x=10^{0}=1\), \(f(x)=\frac{(1)^{2}-1}{0.02*(1)^{2}}=0\). Do this for all \(x\) values.

Record the Calculated Values

For each \(x\), calculate the \(f(x)\) value and record it in the table. If done correctly, the table should look like this: \[ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \ \hline f(x) & 0 & 49 & 4999 & 499999 & 49999999 & 4999999999 & 499999999999 \ \hline \end{array} \]

Estimating the limit as x approaches infinity

Examine the table to estimate the limit of \(f(x)\) as \(x\) approaches infinity. Here, as \(x\) tends towards infinity, the value of \(f(x)\) also increases, looking like it trends towards infinity. However, consider the actual form of the function again, which simplifies to \(f(x)=50-(\frac{50}{x^{2}})\). The second term, \(\frac{50}{x^{2}}\) gets smaller and smaller as \(x\) gets larger and larger (tends to infinity), thereby making \(f(x)\) approach the value of 50 as \(x\) tends to infinity.

Key concepts

Graphing Utility

Graphing utilities are powerful tools in mathematics that help you visualize functions and their behavior over different intervals. These utilities are available in various forms, such as software programs or graphing calculators. Their primary function is to allow you to plot functions quickly. This makes analyzing functions much easier than doing calculations manually.

Using a graphing utility, you can input a range of values for a variable and instantly see the resulting function values. This is extremely useful when dealing with complex functions or when you need to see how a function behaves as the input value becomes very large, as in the given exercise.

• Interactive visual representation: Graphing utilities allow you to interactively see how changes in the input affect the function's output.
• Speed: Quickly plot and calculate function values without manual effort.
• Accuracy: Helps ensure precision in calculations by reducing human error.

When estimating the limit of a function as its input approaches a certain value, graphing utilities can effectively demonstrate trends and asymptotic behavior.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. They take the general form of:\[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\]where each \(a\) is a coefficient, and \(n\) is a non-negative integer indicating the power of \(x\). The degrees of these powers determine many characteristics of the polynomial's graph, including its general shape and number of turns.

In this exercise, the function \(f(x) = \frac{x^2 - 1}{0.02 x^2}\) is a rational function derived from polynomials in both the numerator and the denominator. When analyzing polynomials, particular attention is paid to the leading term as \(x\) increases or decreases substantially since it dominates the behavior of the function.

• Order influence: The highest power determines the end behavior of the polynomial.
• Roots: The solutions of the polynomial equation (where \(f(x)=0\)).
• Behavior as \(x\) approaches infinity: In large limits, leading terms primarily affect growth rates.

Understanding polynomial functions helps in predicting function behavior and identifying limits at infinity.

Infinity

Infinity is a concept in mathematics that is used to describe something that is unbounded or limitless. It is not a number in the traditional sense, but rather an idea that helps us understand growth and limits in various mathematical contexts.

When calculating limits, especially as the variable \(x\) approaches infinity, it is important to focus on how the function behaves rather than its exact values. In the case of the function \(f(x) = \frac{x^2 - 1}{0.02 x^2}\), it's useful to simplify it first:\[f(x) = 50 - \frac{50}{x^2}\]The negative quotient \(\frac{50}{x^2}\) gets closer to zero as \(x\) grows larger, and thus \(f(x)\) gets closer to a value of 50.

• Concept of infinity: It indicates unending growth or decline.
• Limits at infinity: These help find the function's behavior as input becomes very large.
• Dominance of terms: Leading terms in a polynomial can determine behavior as \(x\) approaches infinity.

Understanding infinity in terms of limits and function behavior is crucial for analyzing and interpreting mathematical scenarios.