Question

the function ( f ) is given by ( f(x)=5x^{4}-2x^{3}-3 ). which of the following describes the end - behavior of ( f )?

Explanation:

Step1: Identify the leading - term

The function is \(f(x)=5x^{4}-2x^{3}-3\). The leading - term is \(5x^{4}\) since it has the highest degree.

Step2: Analyze the end - behavior based on the leading - term

For a polynomial function \(y = ax^{n}\), when \(n\) is even and \(a>0\), \(\lim_{x
ightarrow-\infty}ax^{n}=\infty\) and \(\lim_{x
ightarrow\infty}ax^{n}=\infty\). Here, \(n = 4\) (even) and \(a = 5>0\). So, \(\lim_{x
ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=\infty\).

Answer:

\(\lim_{x
ightarrow-\infty}f(x)=\infty\) and \(\lim_{x
ightarrow\infty}f(x)=\infty\)