Questions: Describe the long run behavior of f(t)=-5t^9-3t^6+4t^5+1 As t → -∞, f(t) → As t → ∞, f(t) →

Solution Steps

To determine the long-run behavior of the polynomial function \( f(t) = -5t^9 - 3t^6 + 4t^5 + 1 \), we focus on the term with the highest degree, which is \(-5t^9\). As \( t \rightarrow \infty \) or \( t \rightarrow -\infty \), the behavior of the polynomial is dominated by this term. Since the coefficient of \( t^9 \) is negative, as \( t \rightarrow \infty \), \( f(t) \rightarrow -\infty \) and as \( t \rightarrow -\infty \), \( f(t) \rightarrow \infty \).

The function given is \( f(t) = -5t^9 - 3t^6 + 4t^5 + 1 \). To determine the long-run behavior as \( t \) approaches positive and negative infinity, we focus on the term with the highest degree, which is \( -5t^9 \).

As \( t \rightarrow \infty \), the dominant term \( -5t^9 \) dictates the behavior of the function. Therefore, we find: \[ \lim_{t \to \infty} f(t) = -\infty \]

Similarly, as \( t \rightarrow -\infty \), the term \( -5t^9 \) still dominates. Since \( t^9 \) is negative for negative \( t \), we have: \[ \lim_{t \to -\infty} f(t) = \infty \]

Final Answer