Problem 34 - Calculus for Biology and Medicine

Question

Use a table and a graph to find out what happens to $$ f(x)=\frac{2 x}{x-1} $$ as \(x \rightarrow \infty\). What happens as \(x \rightarrow-\infty\) ? What happens as \(x \rightarrow 1 ?\)

Step-by-Step Solution

Answer

As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 2 \). As \( x \to 1 \), \( f(x) \to \infty \) or \( f(x) \to -\infty \).

Step 1: Create a Table of Values

Evaluate the function \( f(x) = \frac{2x}{x-1} \) for several positive and negative values of \( x \). This will help us observe the behavior of \( f(x) \) as \( x \) approaches certain limits, specifically \( \infty, -\infty, \) and \( 1 \). For example:- As \( x \) approaches \( \infty \), try \( x = 10, 100, 1000 \).- As \( x \) approaches \( -\infty \), try \( x = -10, -100, -1000 \).- As \( x \) approaches \( 1 \), try values very close to 1 like \( x = 0.9, 0.99, 1.1, 1.01 \).Calculate the corresponding \( f(x) \) values for each \( x \).

Step 2: Analyze Table Results for Limits

From the table, observe the values of \( f(x) \):- As \( x \rightarrow \infty \), you will see that \( f(x) \) approaches 2, indicated by the values converging towards 2.- As \( x \rightarrow -\infty \), you also see that \( f(x) \) approaches 2 from the negative side.- As \( x \rightarrow 1 \), the function \( f(x) \) becomes undefined as \( x-1 \to 0 \), causing the values to approach \( \pm \infty \).

Step 3: Graph the Function

Plot the function \( f(x) = \frac{2x}{x-1} \) on a graph for a range that includes \( x \rightarrow \infty, x \rightarrow -\infty, \) and \( x \approx 1 \).Check that:- The graph shows a horizontal asymptote at \( y = 2 \) as \( x \to \infty \) and \( x \to -\infty \).- A vertical asymptote at \( x = 1 \) shows that \( f(x) \) is undefined at this point, with the graph diverging to \( \pm \infty \).

Step 4: Conclude Behavior of the Function

From the table and the graph, conclude the behavior:- As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 2 \).- As \( x \to 1 \), \( f(x) \to \infty \) or \( f(x) \to -\infty \), with a vertical asymptote at \( x = 1 \).

Key Concepts

Understanding Asymptotes

When exploring limits in calculus, asymptotes are crucial in understanding the behavior of functions as they approach certain values. An asymptote is a line that a graph approaches but never quite reaches. There are two main types of asymptotes: vertical and horizontal.

• Vertical asymptotes occur where a function becomes undefined, often caused by division by zero. For the function \( f(x) = \frac{2x}{x-1} \), a vertical asymptote is at \(x = 1\), because the denominator becomes zero. Here, the function heads toward \( \pm \infty \) as \(x\) nears 1.
• Horizontal asymptotes describe where a function heads as \(x\) moves towards infinity in either direction. For our function, the horizontal asymptote is \( y = 2 \). This indicates that as \(x\) either goes to positive or negative infinity, \( f(x) \) approaches the value of 2.

Understanding these lines can clarify how a graph behaves near significant points and aid in predicting behaviors over large values of \(x\).

Exploring Function Behavior

The behavior of a function gives us insight into how its values change with varying\( x \). This means determining how the function behaves as \( x \) approaches certain critical points.

• As \( x \to \infty \), \( f(x) = \frac{2x}{x-1} \) approaches 2. This describes the function leveling off to a horizontal asymptote.
• Similarly, as \( x \to -\infty \), \( f(x) \) also approximates 2, showing consistent behavior regardless of whether \(x\) is highly positive or highly negative.
• Things change as \( x \to 1 \). Here, \( f(x) \) grows without bound to \( \pm \infty \), due to the vertical asymptote at \( x = 1 \). This is depicted as the graph shooting upwards or downwards rapidly.

Each behavior tells a different part of the story of how this function interacts with its input values and provides a comprehensive view when understood collectively.

Utilizing a Table of Values

A table of values is a helpful tool in visualizing how function values change as you modify \( x \). By calculating \( f(x) = \frac{2x}{x-1} \) for various \( x \):

• Choose large positive numbers to see how \( f(x) \) behaves as \( x \to \infty \).
• Try large negative numbers to understand the function as \( x \to -\infty \).
• Observe values close to 1 for insight on \( f(x) \)'s behavior as it approaches the vertical asymptote.

For instance, if you substitute \( x = 10, 100, 1000 \), you'll notice \( f(x) \) values nearing 2. Conversely, substitute \( x = -10, -100, -1000 \) for similar convergence from the negative end. By doing so, patterns emerge, aiding in understanding the function's tendencies across its domain.

Graphing Functions for Visual Representation

Graphing functions provides a clear, visual method to understand the complex behavior of equations like \( f(x) = \frac{2x}{x-1} \). A graph gives a snapshot of everything:

• Displays horizontal asymptotes such as \( y = 2 \), perceptible as a line the graph edges closer to as \( x \to \pm \infty \).
• Reveals the vertical asymptote at \( x = 1 \), showing where the function is undefined with the line skyrocketing toward infinity.
• Indicates where the function moves as a whole across \( x \)'s scale, allowing an intuitive understanding of its overall trend.

By plotting crucial points and observing these intersections with asymptotes, graphs illustrate a function's behavior over various ranges, cementing both the limits and the infinite behaviors in a way numbers alone might not.