Question
Describe the long run behavior of f(x)=x^6
As x → - oo, f(x) → oo, -oo, 0?
As x → oo, f(x) → oo, -oo, 0
As x → - oo, f(x) → oo, -oo, 0?
As x → oo, f(x) → oo, -oo, 0
Answers
Answered by
GPT-4o mini
To analyze the long-run behavior of the function \( f(x) = x^6 \) as \( x \) approaches negative infinity and positive infinity, we can evaluate the limits directly.
1. **As \( x \to -\infty \)**:
When \( x \) is a very large negative number, raising it to an even power (such as 6) will result in a positive value. Specifically, we can express this mathematically:
\[
\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} x^6 = \infty
\]
Thus, as \( x \to -\infty \), \( f(x) \to \infty \).
2. **As \( x \to \infty \)**:
When \( x \) is a very large positive number, raising it to the sixth power will still yield a positive value:
\[
\lim_{x \to \infty} f(x) = \lim_{x \to \infty} x^6 = \infty
\]
Therefore, as \( x \to \infty \), \( f(x) \to \infty \).
In summary:
- As \( x \to -\infty \), \( f(x) \to \infty \).
- As \( x \to \infty \), \( f(x) \to \infty \).
1. **As \( x \to -\infty \)**:
When \( x \) is a very large negative number, raising it to an even power (such as 6) will result in a positive value. Specifically, we can express this mathematically:
\[
\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} x^6 = \infty
\]
Thus, as \( x \to -\infty \), \( f(x) \to \infty \).
2. **As \( x \to \infty \)**:
When \( x \) is a very large positive number, raising it to the sixth power will still yield a positive value:
\[
\lim_{x \to \infty} f(x) = \lim_{x \to \infty} x^6 = \infty
\]
Therefore, as \( x \to \infty \), \( f(x) \to \infty \).
In summary:
- As \( x \to -\infty \), \( f(x) \to \infty \).
- As \( x \to \infty \), \( f(x) \to \infty \).