Question

the function f(x) is given by $f(x)=2x^{6}+5x^{4}+6x + 1$. which of the following correctly describes the end - behavior of f as the input values increase without bound? $lim_{x
ightarrowinfty}f(x)=-infty$ $lim_{x
ightarrowinfty}f(x)=infty$ $lim_{x
ightarrow-infty}f(x)=-infty$ $lim_{x
ightarrow-infty}f(x)=infty$

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function $f(x)=2x^{6}+5x^{4}+6x + 1$ is $2x^{6}$.

Step2: Analyze the end - behavior for $x\to\infty$

For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ even and $a_n>0$, as $x\to\infty$, $y\to\infty$. Here, $n = 6$ (even) and $a_n=2>0$, so $\lim_{x\to\infty}f(x)=\infty$.

Step3: Analyze the end - behavior for $x\to-\infty$

For a polynomial $y = a_nx^n+\cdots+a_0$ with $n$ even and $a_n>0$, as $x\to-\infty$, $y\to\infty$. Since $n = 6$ (even) and $a_n = 2>0$, $\lim_{x\to-\infty}f(x)=\infty$.

Answer:

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