Problem 53 - Precalculus with Limits

Question

OXYGEN LEVEL Suppose that \(f(t)\) measures the level of oxygen in a pond, where \(f(t) = 1\) is the normal (unpolluted) level and the time \(t\) is measured in weeks. When \(t=0\), organic waste is dumped into the pond, and as the waste material oxidizes, the level of oxygen in the pond is given by $$ f(t) = \dfrac{t^2 - t + 1}{t^2 + 1} $$. (a) What is the limit of \(f\) as \(t\) approaches infinity? (b) Use a graphing utility to graph the function and verify the result of part (a). (c) Explain the meaning of the limit in the context of the problem.

Step-by-Step Solution

Answer

The limit of the function as \(t\) approaches infinity is 1. Graphically, this is confirmed by the plot of the function \(f(t) = \dfrac{t^2 - t + 1}{t^2 + 1}\), which shows the function approaching the value 1 as \(t\) approaches infinity. In the context of the problem, this limit indicates that as time progresses, the oxygen level in the pond returns to its normal, unpolluted level of 1.

Step 1: Compute the limit of the function as \(t\) approaches infinity

To find the limit of the function \(f(t) = \dfrac{t^2 - t + 1}{t^2 + 1}\) as \(t\) approaches infinity, we need to look at the leading terms which are \(t^2\) in both the numerator and the denominator. When \(t\) grows very large, the term \(-t\) in the numerator and the constant \(1\) in the numerator and denominator become negligible. So, we can say that the limit of \(f(t)\) as \(t\) approaches infinity is \(1\). That is, \(\lim_{{t \to \infty}} f(t) = 1\).

Step 2: Graph the function using a graphing utility

Plotting the function \(f(t) = \dfrac{t^2 - t + 1}{t^2 + 1}\) using a suitable graphing utility will provide a visual confirmation of the result obtained in Step 1. The graph should show the function approaching a value of 1 as \(t\) approaches infinity.

Step 3: Explain the meaning of the limit in the context of the problem

The limit found tells us that as time \(t\) goes on indefinitely (approaches infinity), the function \(f(t)\), which represents the level of oxygen in the pond, approaches 1. Interpretively, this indicates that over an extended period, the level of oxygen in the pond will return to its normal (unpolluted) level of 1 as the organic waste material oxidizes completely.

Key Concepts

Rational Functions

Rational functions are expressions that involve the ratio of two polynomials. In these functions, both the numerator and the denominator have polynomial expressions. For example, the function given in the exercise, \[ f(t) = \dfrac{t^2 - t + 1}{t^2 + 1} \]is a rational function. Here, both the numerator and denominator are polynomials in terms of the variable \( t \). It is important to consider the degrees of these polynomials because they often determine the behavior of the function as \( t \) approaches infinity.

• The degree of a polynomial is the highest power of the variable present in the expression.
• A rational function's behavior as the input grows large can often be predicted by comparing the leading terms—the terms with the highest powers—in the numerator and the denominator.

Analyzing rational functions at the extremes often helps in understanding more complex problems like the oxygen levels in our given exercise.

Asymptotic Behavior

Asymptotic behavior in mathematics refers to how a function behaves as the input approaches a certain value, often infinity. For rational functions, it’s particularly useful to investigate their horizontal asymptotes. These horizontal asymptotes help us understand the end behavior of the function.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In the oxygen level exercise, since the degrees of the polynomials in the numerator and the denominator are both \( 2 \), the horizontal asymptote is at \( y = \frac{1}{1} = 1 \). This indicates that as time goes on indefinitely (asymptotically), the oxygen level in the pond returns to the unpolluted level.

Graphical Analysis

Graphical analysis of functions involves using graphs to visually interpret the behavior of functions over their domain. For the function \( f(t) = \dfrac{t^2 - t + 1}{t^2 + 1} \), drawing its graph can provide tangible insights into how the function behaves as \( t \) increases.
An important part of graphical analysis is looking for:

• Intercepts, which tell us where the function crosses the axes.
• Asymptotic lines, which indicate the function's long-term behavior.
• Increasing or decreasing intervals that show where the function is rising or falling.

When graphing \( f(t) \), we notice that as \( t \) approaches infinity, the function approaches the horizontal line \( y = 1 \). This graphical representation confirms the result from our asymptotic analysis, as it clearly shows that the oxygen level stabilizes at 1, the original unpolluted level, after some time. Graphs are invaluable for visualizing complex functions and verifying theoretical findings.