Question

each function using limit notation.

1. $p(x) = 7x^4 + 3x^3 - 3x - 4$

Explanation:

Step1: Identify end behavior rules

For polynomial $p(x)=a_nx^n+...+a_0$, end behavior depends on leading term $a_nx^n$.

Step2: Define leading term of $p(x)$

Leading term: $7x^4$, where $a_n=7>0$, $n=4$ (even).

Step3: Apply limit for $x\to+\infty$

As $x\to+\infty$, $x^4\to+\infty$, so $7x^4\to+\infty$.
$\lim_{x\to+\infty} p(x) = \lim_{x\to+\infty} (7x^4+3x^3-3x-4) = +\infty$

Step4: Apply limit for $x\to-\infty$

As $x\to-\infty$, $x^4=(-|x|)^4=|x|^4\to+\infty$, so $7x^4\to+\infty$.
$\lim_{x\to-\infty} p(x) = \lim_{x\to-\infty} (7x^4+3x^3-3x-4) = +\infty$

Answer:

$\lim_{x\to+\infty} p(x) = +\infty$ and $\lim_{x\to-\infty} p(x) = +\infty$