Questions: Use limit notation to describe the unbounded behavior of the given function as (x) approaches (infty) and as (x) approaches (-infty). [ k(x)=-2.4 x^5+5.3 x-0.2 ] Answer [ lim x rightarrow infty k(x)=square ] [ lim x rightarrow-infty k(x)=square ]

Solution Steps

To determine the unbounded behavior of the function \( k(x) = -2.4x^5 + 5.3x - 0.2 \) as \( x \) approaches \( \infty \) and \( -\infty \), we need to analyze the leading term of the polynomial, which is \( -2.4x^5 \). The leading term will dominate the behavior of the function for large values of \( x \).

1. As \( x \) approaches \( \infty \), the leading term \( -2.4x^5 \) will tend to \( -\infty \) because the coefficient is negative.
2. As \( x \) approaches \( -\infty \), the leading term \( -2.4x^5 \) will tend to \( \infty \) because raising a negative number to an odd power results in a negative value, and the negative coefficient will make it positive.

To determine the unbounded behavior of the function \( k(x) = -2.4x^5 + 5.3x - 0.2 \) as \( x \) approaches \( \infty \) and \( -\infty \), we focus on the leading term \( -2.4x^5 \). This term will dominate the behavior of the function for large values of \( x \).

As \( x \) approaches \( \infty \), the leading term \( -2.4x^5 \) will tend to \( -\infty \) because the coefficient is negative. Therefore, \[ \lim_{x \to \infty} k(x) = -\infty \]

As \( x \) approaches \( -\infty \), the leading term \( -2.4x^5 \) will tend to \( \infty \) because raising a negative number to an odd power results in a negative value, and the negative coefficient will make it positive. Therefore, \[ \lim_{x \to -\infty} k(x) = \infty \]

Final Answer