Describe the end behavior of the following function using limit notation h(x)=-x” +5×2-19
The Correct Answer and Explanation is :
To describe the end behavior of the function ( h(x) = -x^3 + 5x^2 – 19 ), we analyze how the function behaves as ( x ) approaches positive and negative infinity. In other words, we want to find the limits of the function as ( x ) gets very large (positively and negatively).
End Behavior:
- As ( x \to \infty ) (as ( x ) becomes very large): The function ( h(x) ) is a cubic polynomial, so we focus on the highest degree term for the behavior at large values of ( x ). The highest degree term in ( h(x) ) is ( -x^3 ), since the cubic term will dominate as ( x ) becomes large. [
\lim_{x \to \infty} h(x) = \lim_{x \to \infty} (-x^3 + 5x^2 – 19)
] As ( x \to \infty ), ( -x^3 ) becomes very negative, overpowering the other terms. So, [
\lim_{x \to \infty} h(x) = -\infty
] - As ( x \to -\infty ) (as ( x ) becomes very negative): Similarly, we consider the behavior as ( x \to -\infty ). The cubic term ( -x^3 ) still dominates, but since ( x^3 ) is negative when ( x ) is negative, ( -x^3 ) will become very positive. Therefore, the limit is: [
\lim_{x \to -\infty} h(x) = \lim_{x \to -\infty} (-x^3 + 5x^2 – 19)
] As ( x \to -\infty ), the term ( -x^3 ) grows large positively (since a negative raised to an odd power gives a negative, and multiplying by a negative flips it), and this causes the function to tend towards positive infinity. Thus, [
\lim_{x \to -\infty} h(x) = \infty
]
Conclusion:
- The end behavior of the function is: [
\lim_{x \to \infty} h(x) = -\infty
]
[
\lim_{x \to -\infty} h(x) = \infty
]
This means that as ( x ) increases without bound, the function decreases without bound (approaches negative infinity), and as ( x ) decreases without bound, the function increases without bound (approaches positive infinity).