the function $f$ is given by $f(x)=5x^{6}-2x^{3}-3$. which of the following describes the end - behavior of…

the function $f$ is given by $f(x)=5x^{6}-2x^{3}-3$. which of the following describes the end - behavior of $f$?\n(a) $lim_{x\rightarrow-infty}f(x)=-infty$ and $lim_{x\rightarrowinfty}f(x)=-infty$\n(b) $lim_{x\rightarrow-infty}f(x)=infty$ and $lim_{x\rightarrowinfty}f(x)=infty$\n(c) $lim_{x\rightarrow-infty}f(x)=-infty$ and $lim_{x\rightarrowinfty}f(x)=infty$\n(d) $lim_{x\rightarrow-infty}f(x)=infty$ and $lim_{x\rightarrowinfty}f(x)=-infty$

the function $f$ is given by $f(x)=5x^{6}-2x^{3}-3$. which of the following describes the end - behavior of $f$?\n(a) $lim_{x\rightarrow-infty}f(x)=-infty$ and $lim_{x\rightarrowinfty}f(x)=-infty$\n(b) $lim_{x\rightarrow-infty}f(x)=infty$ and $lim_{x\rightarrowinfty}f(x)=infty$\n(c) $lim_{x\rightarrow-infty}f(x)=-infty$ and $lim_{x\rightarrowinfty}f(x)=infty$\n(d) $lim_{x\rightarrow-infty}f(x)=infty$ and $lim_{x\rightarrowinfty}f(x)=-infty$

Answer

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function $f(x)=5x^{6}-2x^{3}-3$ is $5x^{6}$.

Step2: Analyze the degree and leading - coefficient

The degree of the leading - term $n = 6$ (an even number), and the leading - coefficient $a = 5$ (a positive number). For a polynomial function $y = a_nx^n+\cdots+a_0$ with $n$ even and $a_n>0$, we have $\lim_{x\rightarrow-\infty}y=\infty$ and $\lim_{x\rightarrow\infty}y=\infty$. As $x\rightarrow-\infty$, $f(x)=5x^{6}-2x^{3}-3\approx5x^{6}\rightarrow\infty$ since $x^{6}>0$ for all $x\neq0$. As $x\rightarrow\infty$, $f(x)=5x^{6}-2x^{3}-3\approx5x^{6}\rightarrow\infty$.

Answer:

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