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

Transcript text: Describe the long run behavior of $f(x)=-5 x^{8}+5 x^{7}+4 x^{2}-4$ As $x \rightarrow-\infty, f(x) \rightarrow ?$ As $x \rightarrow \infty, f(x) \rightarrow ?$

Solution

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 $.

As $ x \rightarrow \infty $, the leading term $ -5x^8 $ will dominate, and since it is negative, $ f(x) $ will approach $ -\infty $.
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 $.

Step 1: Analyze the Function

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 $.

Step 2: Evaluate the Limit as $ x \rightarrow \infty $

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

Step 3: Evaluate the Limit as $ x \rightarrow -\infty $

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