Problem 289 - Calculus Volume 1

Question

For the following exercises, graph the function on a graphing calculator on the window \(x=[-5,5]\) and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit. $$ f(x)=\frac{1}{x+10} $$

Step-by-Step Solution

Answer

The horizontal asymptote of \( f(x) = \frac{1}{x+10} \) is \( y = 0 \).

Step 1: Graph the Function

First, enter the function \(f(x) = \frac{1}{x+10}\) into your graphing calculator. Set the window range for \(x\) between \(-5\) and \(5\). Observe the graph carefully to get an initial idea of its behavior as \(x\) moves towards positive and negative infinity.

Step 2: Estimate the Horizontal Asymptote

Based on the graph in the given window, note the behavior of the function values as \(x\) moves to the far left and right. Observe that the values of \(f(x)\) approach a certain horizontal line as they extend beyond the visible region. This suggests a horizontal asymptote.

Step 3: Define and Calculate the Horizontal Asymptote

For a function \(f(x) = \frac{1}{ax+b}\), as \(x\to\pm\infty\), the value of the function approaches \(0\) because the constant term \(b\) in the numerator remains fixed while the denominator grows infinitely large. Thus, the horizontal asymptote is \(y = 0\).

Step 4: Verify the Asymptote with Limits

Calculate \( \lim_{{x \to \infty}} \frac{1}{x+10} \) and \( \lim_{{x \to -\infty}} \frac{1}{x+10} \). Both limits evaluate to \(0\), confirming that the horizontal asymptote is indeed \(y = 0\).

Key Concepts

Understanding Graphing Calculators

Graphing calculators are powerful tools that allow us to visualize the behavior of functions. When tackling exercises that involve functions and their limits, graphing calculators can be indispensable.
By graphing the function, like in this exercise where we have the function \(f(x) = \frac{1}{x+10}\), we set the x-axis window from \([-5,5]\). This specific window helps us focus on a particular section of the graph. Although the full behavior might not be visible in this range, it gives us a starting point to observe trends and patterns.
By plotting the graph, we can visually identify where the function seems to approach a horizontal line as \(x\) extends towards large positive or negative values. This "approaching" behavior offers insight into where potential horizontal asymptotes may appear.
Using a graphing calculator effectively means regularly adjusting your view and zooming in or out as needed. Keep in mind that even slight window adjustments might reveal more about the function's behavior.

Analyzing Limit Calculations

Limit calculations are crucial for finding where a function might have a horizontal asymptote. A horizontal asymptote is a horizontal line that the graph of a function approaches as \(x\) goes to infinity or negative infinity.
The function \(f(x) = \frac{1}{x+10}\) approaches 0 as \(x\) becomes very large in either direction. Mathematically, this is expressed using limits. Calculate \( \lim_{{x \to \infty}} \frac{1}{x+10} = 0 \) and \( \lim_{{x \to -\infty}} \frac{1}{x+10} = 0 \).
The numerator here is a constant 1, but as \(x\) increases or decreases without bound, the denominator \(x+10\) also grows infinitely. As the denominator becomes very large, the fraction tends to zero, explaining the function's behavior and confirming the horizontal asymptote \(y = 0\).

Interpreting Function Behavior

Understanding the behavior of a function helps in predicting how it will act over its domain. Functions like \(f(x) = \frac{1}{x+10}\) exhibit consistent patterns that are typical for rational functions with linear denominators.
As you plot these functions, you’ll often see them approach but never actually reach certain lines, indicating asymptotic behavior. This is because the function's value changes very slowly and might even appear to level out. However, it never quite reaches the asymptotic line.
By noticing these trends—like \(f(x)\) getting closer and closer to a specific value without touching it—you can discern where the asymptotes are. Typically, rational functions will have horizontal asymptotes if the degree of the denominator is greater than the degree of the numerator, as we see here with the horizontal asymptote at \(y = 0\).
Understanding these patterns helps not only in graphing but also in anticipating function behavior across different mathematical problems.