Verified Answer :

  • Expand the denominator of the function \(f(x) = \frac{x-2}{(x-3)^2}\) to get \(f(x) = \frac{x-2}{x^2 - 6x + 9}\).
  • Calculate the limit of the function as \(x\) approaches infinity by dividing the numerator and denominator by the highest power of \(x\), which is \(x^2\).
  • Observe that as \(x \to \infty\), the terms \(\frac{1}{x}\) and \(\frac{1}{x^2}\) approach zero, leaving a limit of \(\frac{0}{1} = 0\).
  • Identify the horizontal asymptote as \(\boxed{y=0}\).

Explanation

  1. Problem Analysis and Goal Definition

We are given the rational function \(f(x) = \frac{x-2}{(x-3)^2}\) and are tasked with finding its horizontal asymptote. A horizontal asymptote is a horizontal line \(y = L\) that the graph of a function approaches as \(x\) approaches positive or negative infinity. To find this, we evaluate the limit of the function as \(x \to \infty\) and \(x \to -\infty\).

  1. Expanding the Function and Analyzing the Limit

First, let us expand the denominator \((x-3)^2\) to make the function easier to analyze:
\[(x-3)^2 = x^2 - 6x + 9\]
So, the function can be written as:
\[f(x) = \frac{x-2}{x^2 - 6x + 9}\]
Now, we examine the behavior of \(f(x)\) as \(x\) becomes very large. We can divide both the numerator and the denominator by the highest power of \(x\) present in the denominator, which is \(x^2\):
\[\lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{\frac{x}{x^2} - \frac{2}{x^2}}{\frac{x^2}{x^2} - \frac{6x}{x^2} + \frac{9}{x^2}} = \lim_{x \to \infty} \frac{\frac{1}{x} - \frac{2}{x^2}}{1 - \frac{6}{x} + \frac{9}{x^2}}\]

  1. Calculating the Limit and Determining the Asymptote

As \(x\) approaches infinity, the terms \(\frac{1}{x}\), \(\frac{2}{x^2}\), \(\frac{6}{x}\), and \(\frac{9}{x^2}\) all approach zero. Substituting these values into our expression gives:
\[\lim_{x \to \infty} \frac{0 - 0}{1 - 0 + 0} = \frac{0}{1} = 0\]
The same result is obtained for the limit as \(x \to -\infty\). Since the limit of the function as \(x \to \pm \infty\) is \(0\), the horizontal asymptote of the function is \(y = 0\). Comparing this to the given options, we select \(y=0\).

Examples

In data science and engineering, identifying horizontal asymptotes is crucial for understanding the long-term behavior of a system. For example, if a model represents the concentration of a chemical in a reaction over time, the horizontal asymptote shows the steady-state concentration reached after a very long time, allowing scientists to predict the system's final equilibrium state. Knowing how to calculate this limit ensures that predictions about system stability are mathematically sound and reliable.

Related Questions