Questions: Describe the long run behavior of f(x)=-5x^8+5x^7+4x^2-4 As x → -∞, f(x) → ? As x → ∞, f(x) → ?

Solution Steps

To determine the long run behavior of the polynomial function $f(x) = -5x^8 + 5x^7 + 4x^2 - 4$, we need to analyze the leading term, which is $-5x^8$. The leading term will dominate the behavior of the function as $x$ approaches $\infty$ or $-\infty$.

1. As $x \rightarrow \infty$, the leading term $-5x^8$ will dominate, and since it is negative, $f(x)$ will approach $-\infty$.
2. As $x \rightarrow -\infty$, the leading term $-5x^8$ will also dominate, and since $(-x)^8$ is positive and multiplied by a negative coefficient, $f(x)$ will again approach $-\infty$.

The function given is $f(x) = -5x^8 + 5x^7 + 4x^2 - 4$. To determine the long run behavior, we focus on the leading term, which is $-5x^8$. This term will dominate the function as $x$ approaches $\infty$ and $-\infty$.

As $x \rightarrow \infty$: $$f(x) \approx -5x^8 \rightarrow -\infty$$ Thus, we conclude that: $$\lim_{x \to \infty} f(x) = -\infty$$

As $x \rightarrow -\infty$: $$f(x) \approx -5x^8 \rightarrow -\infty$$ Thus, we conclude that: $$\lim_{x \to -\infty} f(x) = -\infty$$

Final Answer