Question

1. describe the end - behavior of the polynomial function $f(x)=-2x^{4}+2x^{3}+4x^{2}$ using correct limit notation.

Explanation:

Step1: Recall end - behavior rule for polynomials

For a polynomial $f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\cdots+a_0$, the end - behavior is determined by the leading term $a_nx^n$. Here, the leading term of $f(x)=-2x^4 + 2x^3+4x^2$ is $-2x^4$ with $n = 4$ (even) and $a_n=-2<0$.

Step2: Find limit as $x\to+\infty$

We use the limit notation $\lim_{x\to+\infty}f(x)$. Since the leading term dominates as $x$ gets large, $\lim_{x\to+\infty}(-2x^4 + 2x^3+4x^2)=\lim_{x\to+\infty}(-2x^4)=-\infty$ because a negative coefficient times a very large positive number (raised to an even power) is a very large negative number.

Step3: Find limit as $x\to-\infty$

We use the limit notation $\lim_{x\to-\infty}f(x)$. Since $n = 4$ (even), $(-x)^4=x^4$. So $\lim_{x\to-\infty}(-2x^4 + 2x^3+4x^2)=\lim_{x\to-\infty}(-2x^4)=-\infty$ because a negative coefficient times a very large positive number (since $(-x)^4=x^4$ for even $n$) is a very large negative number.

Answer:

$\lim_{x\to+\infty}(-2x^4 + 2x^3+4x^2)=-\infty$, $\lim_{x\to-\infty}(-2x^4 + 2x^3+4x^2)=-\infty$