Question

Describe the end behavior (long run behavior) of f(x)=-x^8
As x → -oo, f(x) → oo, -oo, 0

As x,oo, f(x) → oo, -oo, 0
,

Answer 1

To analyze the end behavior of the function \( f(x) = -x^8 \), we need to consider the limits of \( f(x) \) as \( x \) approaches negative infinity (\( -\infty \)) and positive infinity (\( +\infty \)).

1. **As \( x \to -\infty \):**
- The term \( -x^8 \) will dominate. Since \( x^8 \) is a positive term (because any even power of a negative number is positive), \( -x^8 \) will be negative.
- Therefore, as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity.
- In summary:
\[
\text{As } x \to -\infty, \; f(x) \to -\infty.
\]

2. **As \( x \to +\infty \):**
- Again, the term \( -x^8 \) will dominate, but since \( x^8 \) is positive for positive \( x \), \( -x^8 \) will also be negative.
- Therefore, as \( x \) approaches positive infinity, \( f(x) \) also approaches negative infinity.
- In summary:
\[
\text{As } x \to +\infty, \; f(x) \to -\infty.
\]

### Conclusion:
- As \( x \to -\infty, \, f(x) \to -\infty \).
- As \( x \to +\infty, \, f(x) \to -\infty \).

Thus, the complete end behavior is:
- As \( x \to -\infty, f(x) \to -\infty \)
- As \( x \to +\infty, f(x) \to -\infty \)