- Determine the asymptote of the shared rational component \(\frac{x^2+x+1}{x-4}\) using polynomial division to find \(y = x + 5\).
- Multiply by the respective constants \(\frac{1}{6}\) and \(\frac{1}{8}\) to find the oblique asymptotes: \(y = \frac{1}{6}x + \frac{5}{6}\) and \(y = \frac{1}{8}x + \frac{5}{8}\).
- Compare the slopes \(\frac{1}{6} \approx 0.167\) and \(\frac{1}{8} = 0.125\).
- Conclude that since \(\frac{1}{6} > \frac{1}{8}\), the asymptote for \(f(x)\) is steeper.
- Final Answer: \(\boxed{\text{The oblique asymptote for } f(x) \text{ is steeper than for } g(x).}\)
Explanation
- Finding the Asymptote of the Parent Function
To determine the oblique asymptotes of the functions \(f(x) = \frac{x^2+x+1}{6(x-4)}\) and \(g(x) = \frac{x^2+x+1}{8(x-4)}\), we begin by examining the common structure of both rational functions. An oblique asymptote occurs when the degree of the numerator is one higher than the degree of the denominator. Both functions share the same numerator \(x^2+x+1\) and have a denominator of the form \(k(x-4)\), where \(k\) is a constant. We can find the asymptote by performing polynomial division on the expression \(\frac{x^2+x+1}{x-4}\).
Performing the division:
\[\frac{x^2+x+1}{x-4} = \frac{x(x-4) + 5x + 1}{x-4} = \frac{x(x-4) + 5(x-4) + 20 + 1}{x-4} = x + 5 + \frac{21}{x-4}\]
As \(x \to \infty\), the term \(\frac{21}{x-4}\) approaches \(0\), meaning the oblique asymptote for the parent function \(\frac{x^2+x+1}{x-4}\) is \(y = x + 5\).
- Comparing the Asymptotes
Now, we incorporate the constant factors into the results.
For \(f(x) = \frac{1}{6} \cdot \frac{x^2+x+1}{x-4}\), the oblique asymptote is:
\[y = \frac{1}{6}(x + 5) = \frac{1}{6}x + \frac{5}{6}\]
For \(g(x) = \frac{1}{8} \cdot \frac{x^2+x+1}{x-4}\), the oblique asymptote is:
\[y = \frac{1}{8}(x + 5) = \frac{1}{8}x + \frac{5}{8}\]
Comparing the slopes, we have \(m_f = \frac{1}{6}\) and \(m_g = \frac{1}{8}\). Since \(\frac{1}{6} > \frac{1}{8}\), the slope of the oblique asymptote for \(f(x)\) is greater than the slope of the oblique asymptote for \(g(x)\). A greater slope corresponds to a steeper line in the coordinate plane. Thus, the oblique asymptote for \(f(x)\) is steeper than for \(g(x)\).
- Conclusion
Based on our analysis, we conclude that the oblique asymptote for \(f(x)\) is steeper than the oblique asymptote for \(g(x)\). The final answer is: \(\boxed{\text{The oblique asymptote for } f(x) \text{ is steeper than for } g(x).}\)
Examples
In control engineering, analyzing the long-term behavior of systems is essential. Oblique asymptotes represent the trend of a system's output as its input grows extremely large. By identifying which system's output trends more steeply (has a higher slope), engineers can predict which system will reach higher values faster, allowing for safer and more efficient design of electrical circuits or mechanical stabilizers.