- Identify the leading term of the polynomial \(f(r) = r^8 + 4r^6 - 5r^4 + 3\), which is \(r^8\).
- Recognize that the leading term dictates the behavior of the function as \(|r|\) becomes very large.
- Evaluate the limit for \(r \to \infty\), finding that the function approaches \(\infty\).
- Evaluate the limit for \(r \to -\infty\), noting that the even exponent 8 results in positive values, so the function also approaches \(\infty\).
- Final answers: \(\lim_{r \to -\infty} f(r) = \boxed{\infty}\) and \(\lim_{r \to \infty} f(r) = \boxed{\infty}\).
Explanation
- Polynomial Analysis and Leading Term Identification
The polynomial function is given by \(f(r) = r^8 + 4r^6 - 5r^4 + 3\). To determine the behavior of the function as \(r\) approaches positive or negative infinity, we examine the leading term, which is the term with the highest exponent. In this case, the leading term is \(r^8\). For very large values of \(|r|\), the highest-degree term dominates the behavior of the entire polynomial because it grows much faster than all other terms.
- Limit as \(r \to \infty\)
As \(r \to \infty\), the term \(r^8\) approaches infinity since \(r^8\) grows without bound for large positive values. Because \(r^8\) is the dominant term, the sum \(r^8 + 4r^6 - 5r^4 + 3\) will also approach infinity. We can express this by stating that \(\lim_{r \to \infty} f(r) = \infty\).
- Limit as \(r \to -\infty\)
As \(r \to -\infty\), we consider the behavior of the leading term \(r^8\). Since the exponent 8 is an even number, any negative value raised to the 8th power becomes positive. Therefore, \((-r)^8 = r^8\), and as \(r\) becomes increasingly negative, \(r^8\) still grows without bound in the positive direction. Consequently, the entire polynomial \(f(r)\) also grows without bound, and we conclude that \(\lim_{r \to -\infty} f(r) = \infty\).
- Conclusion of Limits
By analyzing the growth of the leading term \(r^8\), we have determined that the function values increase towards positive infinity as \(r\) goes to either positive or negative infinity. Thus, both limits are \(\infty\). Specifically, \(\lim_{r \to -\infty} f(r) = \boxed{\infty}\) and \(\lim_{r \to \infty} f(r) = \boxed{\infty}\).
Examples
Understanding the end behavior of polynomials is essential in fields like physics and engineering, such as when modeling the energy consumption of a system over time. For instance, if the energy required by a machine is described by a polynomial function \(E(t)\), knowing that \(\lim_{t \to \infty} E(t) = \infty\) warns engineers that the machine will eventually require an unsustainable amount of power, indicating that the model might only be valid for a specific range of time or that the design needs refinement.