describe the end behavior of the function f(x)=3x^2 + 4x^3+15 by finding lim_{x→∞} f(x) and lim_{x→ -∞}…

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function $f(x)=3x^{2}+4x^{3}+15$ is $4x^{3}$ since the degree of $4x^{3}$ (degree 3) is higher than the degrees of $3x^{2}$ (degree 2) and the constant term 15 (degree 0).

Step2: Find $\lim_{x\rightarrow-\infty}f(x)$

For a polynomial function $y = a_nx^n+\cdots+a_0$ with leading - term $a_nx^n$, when $n$ is odd and $a_n>0$, $\lim_{x\rightarrow-\infty}a_nx^n=-\infty$. Here, $n = 3$ (odd) and $a_n = 4>0$. So, $\lim_{x\rightarrow-\infty}(4x^{3}+3x^{2}+15)=-\infty$ because as $x\rightarrow-\infty$, the term $4x^{3}$ dominates the behavior of the function.

Answer:

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