- Identify the leading term of the polynomial \(f(x) = -x^5 - 5x^3 + 12x^2 + 36x\), which is \(-x^5\).
- Analyze the limit as \(x \to -\infty\): since \(n=5\) is odd and the coefficient is negative, the function approaches \(\infty\).
- Analyze the limit as \(x \to \infty\): since \(n=5\) is odd and the coefficient is negative, the function approaches \(-\infty\).
- Conclude that \(\lim_{x \to -\infty} f(x) = \boxed{\infty}\) and \(\lim_{x \to \infty} f(x) = \boxed{-\infty}\).
Explanation
- Analyze the Leading Term
To determine the end behavior of the polynomial \(f(x) = -x^5 - 5x^3 + 12x^2 + 36x\), we examine its highest-degree term, known as the leading term. For any polynomial \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0\), the end behavior as \(x\) approaches \(\pm \infty\) is dominated by the leading term, \(a_n x^n\). In this function, the leading term is \(-x^5\).
- Evaluating the Limit as x approaches negative infinity
The degree of our polynomial is \(n = 5\), which is an odd number, and the leading coefficient is \(a_n = -1\), which is negative. As \(x\) becomes a very large negative number (approaching \(-\infty\)), the term \(x^5\) also becomes a very large negative number. Therefore, \(-x^5\) approaches \(-(-\infty) = \infty\). Since the lower-degree terms in the polynomial are insignificant compared to \(-x^5\) as \(x \to -\infty\), the limit of the entire function is \(\infty\).
- Evaluating the Limit as x approaches positive infinity
As \(x\) becomes a very large positive number (approaching \(\infty\)), the term \(x^5\) grows to \(\infty\). Therefore, the leading term \(-x^5\) approaches \(-(\infty) = -\infty\). Similar to the previous case, the higher-degree term dictates the overall growth, so the limit of the entire function as \(x \to \infty\) is \(-\infty\).
- Final Answer
Based on the analysis of the leading term, we conclude:
\(x \to -\infty, f(x) \to \infty\)
\(x \to \infty, f(x) \to -\infty\)
The final results are \(\boxed{\infty}\) and \(\boxed{-\infty}\), respectively.
Examples
Understanding the end behavior of polynomials is essential in economics and modeling, such as predicting long-term trends for a company's profit function. If a profit function is modeled by a polynomial like \(P(t) = -t^5 + \dots\), knowing that \(P(t) \to -\infty\) as \(t \to \infty\) helps business analysts identify that the current growth model is unsustainable in the long run and must be adjusted before the business faces significant losses. Applying the leading term test allows for quick assessment of these long-term trends without needing to calculate every point of the function.