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

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\rightarrow-\infty}ax^{n}=\infty) and (\lim_{x\rightarrow\infty}ax^{n}=\infty). Here, (n = 4) (even) and (a = 5>0). So, (\lim_{x\rightarrow-\infty}f(x)=\infty) and (\lim_{x\rightarrow\infty}f(x)=\infty).

Answer:

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