Questions: Describe the long run behavior of f(r)=-4r^7+3r^6-2r^5-5 As r → -∞, f(r) → ? As r → ∞, f(r) →

Solution Steps

To determine the long run behavior of the polynomial function \( f(r) = -4r^7 + 3r^6 - 2r^5 - 5 \), we focus on the term with the highest degree, which is \(-4r^7\). As \( r \rightarrow -\infty \) and \( r \rightarrow \infty \), the behavior of the function is dominated by this term. Specifically, since the leading coefficient is negative, the function will tend towards negative infinity in both directions.

The function given is \( f(r) = -4r^7 + 3r^6 - 2r^5 - 5 \). To understand its long run behavior, we focus on the leading term, which is \( -4r^7 \). This term will dominate the behavior of the function as \( r \) approaches positive or negative infinity.

As \( r \) approaches \( -\infty \), the leading term \( -4r^7 \) also approaches \( +\infty \) because raising a negative number to an odd power results in a negative number, and multiplying by \(-4\) makes it positive. Therefore, we conclude that: \[ \lim_{r \to -\infty} f(r) = +\infty \]

As \( r \) approaches \( +\infty \), the leading term \( -4r^7 \) approaches \( -\infty \) since raising a positive number to an odd power results in a positive number, and multiplying by \(-4\) makes it negative. Thus, we find that: \[ \lim_{r \to \infty} f(r) = -\infty \]

Final Answer