Question

Consider the polynomial function f(x)=5x^6-4-6x^4+2x^3+8x^7-8x^8
as x → -oo, f(x) → +oo, -oo, 0
as x → oo, f(x) → +oo, -oo, 0

Answer 1

To analyze the behavior of the polynomial function \( f(x) = 5x^6 - 4 - 6x^4 + 2x^3 + 8x^7 - 8x^8 \) as \( x \to -\infty \) and \( x \to \infty \), we should focus on the leading term of the polynomial since it dominates the behavior of the function for large values of \( |x| \).

### Identify the Leading Term
The leading term in the polynomial \( f(x) = 5x^6 - 4 - 6x^4 + 2x^3 + 8x^7 - 8x^8 \) is \( -8x^8 \). This is because the term with the highest degree (in this case \( 8 \)) will dominate the polynomial's behavior as \( x \) approaches either \( +\infty \) or \( -\infty \).

### Behavior as \( x \to \infty \)
When \( x \to \infty \):
- The leading term \( -8x^8 \) becomes very large and negative, since \( x^8 \) grows very large and negative when multiplied by \(-8\).
- Therefore,
\[
f(x) \to -\infty
\]

### Behavior as \( x \to -\infty \)
When \( x \to -\infty \):
- Again, the leading term \( -8x^8 \) is relevant. Since \( x^8 \) is always positive for real \( x \) (because raising a negative number to an even power yields a positive result), the leading term \( -8x^8 \) still becomes very large and negative as \( x^8 \) increases, leading to
\[
f(x) \to -\infty
\]

### Summary
As a result, we summarize the behavior of \( f(x) \) as follows:
- As \( x \to -\infty \), \( f(x) \to -\infty \).
- As \( x \to \infty \), \( f(x) \to -\infty \).

So the correct interpretations are:
- For \( x \to -\infty, f(x) \to -\infty \).
- For \( x \to +\infty, f(x) \to -\infty \).

The answer options you've provided seem to have inconsistencies with our analysis; hence, the accurate behavior of \( f(x) \) is \( -\infty \) in both limits.