Questions: Describe the long run behavior of f(t)=t^8-4t^6-3t^2+5 As t → -∞, f(t) → ? As t → ∞, f(t) → ?

Solution Steps

To determine the long run behavior of the polynomial function \( f(t) = t^8 - 4t^6 - 3t^2 + 5 \), we focus on the term with the highest degree, which is \( t^8 \). As \( t \rightarrow -\infty \) and \( t \rightarrow \infty \), the behavior of the function is dominated by this term. Since the coefficient of \( t^8 \) is positive, \( f(t) \) will approach \( \infty \) as \( t \rightarrow \infty \) and \( f(t) \) will also approach \( \infty \) as \( t \rightarrow -\infty \).

The function given is \( f(t) = t^8 - 4t^6 - 3t^2 + 5 \). To understand its long run behavior, we focus on the term with the highest degree, which is \( t^8 \).

As \( t \) approaches \( \infty \), the dominant term \( t^8 \) dictates the behavior of the function. Since the coefficient of \( t^8 \) is positive, we have: \[ f(t) \rightarrow \infty \quad \text{as} \quad t \rightarrow \infty \]

Similarly, as \( t \) approaches \( -\infty \), the term \( t^8 \) still dominates because it is an even power. Thus, we also find: \[ f(t) \rightarrow \infty \quad \text{as} \quad t \rightarrow -\infty \]

Final Answer