- For the first function \(\frac{5x^2 - 6}{9x^2 + 20x + 8}\), the degrees of numerator and denominator are equal, so the limit as \(x \rightarrow \pm \infty\) is the ratio of leading coefficients, \(\boxed{\frac{5}{9}}\).
- For the second function \(\frac{16x^3 - 15}{12x + 16}\), the degree of the numerator is higher, leading to dominant growth as \(\frac{4}{3}x^2\), so the limit as \(x \rightarrow \pm \infty\) is \(\boxed{\infty}\).
Explanation
- Limits for the first function
We examine the rational function \(f(x) = \frac{5x^2 - 6}{9x^2 + 20x + 8}\). To find the limits as \(x \rightarrow \pm \infty\), we divide the numerator and the denominator by the highest power of \(x\) in the denominator, which is \(x^2\):
\[f(x) = \frac{\frac{5x^2 - 6}{x^2}}{\frac{9x^2 + 20x + 8}{x^2}} = \frac{5 - \frac{6}{x^2}}{9 + \frac{20}{x} + \frac{8}{x^2}}\]
As \(x \rightarrow \infty\) or \(x \rightarrow -\infty\), the terms \(\frac{6}{x^2}\), \(\frac{20}{x}\), and \(\frac{8}{x^2}\) all approach \(0\). Thus, the limit is:
\[\lim_{x \to \pm \infty} f(x) = \frac{5 - 0}{9 + 0 + 0} = \frac{5}{9}\]
So, as \(x \rightarrow \infty\), \(y \rightarrow \frac{5}{9}\), and as \(x \rightarrow -\infty\), \(y \rightarrow \frac{5}{9}\).
- Limits for the second function
We now examine the rational function \(f(x) = \frac{16x^3 - 15}{12x + 16}\). Here, the degree of the numerator (3) is higher than the degree of the denominator (1). For large \(x\), the function is dominated by the ratio of the highest-degree terms:
\[f(x) \approx \frac{16x^3}{12x} = \frac{4}{3}x^2\]
As \(x \rightarrow \infty\), \(x^2 \rightarrow \infty\), so \(f(x) \rightarrow \infty\).
As \(x \rightarrow -\infty\), \(x^2 \rightarrow \infty\), so \(f(x) \rightarrow \infty\).
Thus, both limits for the second function are \(\infty\).
- Final results summary
Summarizing the findings:
- For \(f(x) = \frac{5x^2 - 6}{9x^2 + 20x + 8}\):
- As \(x \rightarrow \infty\), \(y \rightarrow \frac{5}{9}\)
- As \(x \rightarrow -\infty\), \(y \rightarrow \frac{5}{9}\)
- For \(f(x) = \frac{16x^3 - 15}{12x + 16}\):
- As \(x \rightarrow \infty\), \(y \rightarrow \infty\)
- As \(x \rightarrow -\infty\), \(y \rightarrow \infty\)
Examples
In engineering, analyzing the long-term behavior of a system represented by rational functions is crucial for stability analysis. For example, if the output of a control system is modeled by a rational function, determining the horizontal asymptote (the limit as time or input goes to infinity) tells the engineer what steady-state value the system will reach. If the limit is infinite, the system might be unstable or indicate an uncontrolled growth. Knowing how to calculate these limits allows engineers to design systems that behave predictably over time.