Problem 65 - College Algebra

Question

For the following exercises, make tables to behavior of the function near the vertical asymptote and reffecting the horizontal asymptote $$f(x)=\frac{1}{x-2}$$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 2 \), horizontal asymptote at \( y = 0 \).

Step 1: Identifying the Vertical Asymptote

To find the vertical asymptote of the function \( f(x) = \frac{1}{x-2} \), set the denominator equal to zero: \( x-2 = 0 \). Solving this gives \( x = 2 \). This means there is a vertical asymptote at \( x = 2 \).

Step 2: Table for Vertical Asymptote

Create a table of values approaching the vertical asymptote from both sides. Choose values of \( x \) just smaller and just larger than 2. For example:\[\begin{array}{c|c}x & f(x) \\hline1.9 & 10 \1.99 & 100 \1.999 & 1000 \2.001 & -1000 \2.01 & -100 \2.1 & -10 \\end{array}\]As \( x \to 2^+ \), \( f(x) \to -\infty \) and as \( x \to 2^- \), \( f(x) \to \infty \).

Step 3: Identifying the Horizontal Asymptote

For \( f(x) = \frac{1}{x-2} \), as \( x \to \pm\infty \), the term \( x-2 \to x \) dominates, making \( f(x) \to 0 \). So, the horizontal asymptote is \( y = 0 \).

Step 4: Table for Horizontal Asymptote

Create a table of values for large positive and negative \( x \) to see the approach towards the horizontal asymptote:\[\begin{array}{c|c}x & f(x) \\hline10 & 0.111 \100 & 0.010 \1000 & 0.001 \-10 & -0.111 \-100 & -0.010 \-1000 & -0.001 \\end{array}\]This shows \( f(x) \to 0 \) as \( x \to \pm\infty \).

Key Concepts

Function Behavior

Functions can sometimes behave in fascinating ways as they approach certain values. Understanding how a function behaves helps us identify asymptotes, which act like invisible boundaries.
In the function \( f(x) = \frac{1}{x-2} \), a crucial aspect of its behavior happens near the vertical asymptote at \( x = 2 \). Here, the denominator becomes zero, causing the function to spike dramatically towards either positive or negative infinity.
This dramatic rise or fall means the values of the function become extremely large or small depending on whether you're approaching from the left or the right. On the graph, you'll see that the curve nearly touches the vertical line at \( x = 2 \) but never actually does.

Limits and Approaching Values

Understanding limits helps us predict how a function behaves as it nears specific points or values. It's like knowing what to expect as you get closer to an exit on a highway.
When considering limits for \( f(x) = \frac{1}{x-2} \), we look at what happens as \( x \) approaches 2 from either side:

• As \( x \to 2^+ \), the values of \( f(x) \) head towards \(-\infty \), showing a large negative change.
• As \( x \to 2^- \), \( f(x) \) instead moves towards \(+\infty \), showing a large positive shift.

This distinct separation in behavior is indicative of a vertical asymptote.
Similarly, when looking at horizontal asymptotes, you investigate what happens as \( x \to \pm \infty \). For our function, as \( x \) becomes very large or very small, the effect of the \( -2 \) in the denominator lessens, leading \( f(x) \) to approach 0. Hence, the horizontal asymptote is seen at \( y = 0 \).

Construction of Tables

Tables are a powerful tool in visualizing how a function behaves near its asymptotes. They allow you to see the changes in \( f(x) \) values as \( x \) approaches significant points.
For the function \( f(x) = \frac{1}{x-2} \), constructing a table involves choosing values close to, but not equal to, \( x = 2 \) and observing the changes:

• Values slightly less than 2, such as 1.9, reveal \( f(x) \rightarrow 10 \)
• Closer values like 1.999 push \( f(x) \rightarrow 1000 \)
  showing rapid growth.

You do the same for values slightly greater than 2 to capture the drop towards \(-\infty\), such as 2.001 pushing \( f(x) \rightarrow -1000 \).
For long-term behavior, you use tables with large \( x \) values to see \( f(x) \) nearing 0, reinforcing the horizontal asymptote at \( y=0 \).
This step-by-step table creation helps visualize and confirm theoretical predictions about the function's behavior.