Question

nction using limit notation.

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

$limlimits_{x \to \infty} p(x) = \infty$
$limlimits_{x \to -\infty} p(x) = -\infty$

1. $f(x) = x^3 + 4x^2 - 3$

Explanation:

Step1: Analyze leading term of $p(x)$

For $p(x)=7x^4+3x^3-3x-4$, leading term is $7x^4$, degree 4 (even), leading coefficient $7>0$.

Step2: Find $\lim_{x\to\infty} p(x)$

As $x\to\infty$, $x^4\to\infty$, so $7x^4\to\infty$. Lower degree terms become negligible.
$\lim_{x\to\infty} p(x) = \lim_{x\to\infty}7x^4 = \infty$

Step3: Find $\lim_{x\to-\infty} p(x)$

As $x\to-\infty$, $x^4=(-|x|)^4=|x|^4\to\infty$, so $7x^4\to\infty$. Lower degree terms become negligible.
$\lim_{x\to-\infty} p(x) = \lim_{x\to-\infty}7x^4 = \infty$

Step4: Analyze leading term of $f(x)$

For $f(x)=x^3+4x^2-3$, leading term is $x^3$, degree 3 (odd), leading coefficient $1>0$.

Step5: Find $\lim_{x\to\infty} f(x)$

As $x\to\infty$, $x^3\to\infty$. Lower degree terms become negligible.
$\lim_{x\to\infty} f(x) = \lim_{x\to\infty}x^3 = \infty$

Step6: Find $\lim_{x\to-\infty} f(x)$

As $x\to-\infty$, $x^3=(-|x|)^3=-|x|^3\to-\infty$. Lower degree terms become negligible.
$\lim_{x\to-\infty} f(x) = \lim_{x\to-\infty}x^3 = -\infty$

Answer:

For $p(x)=7x^4+3x^3-3x-4$:
$\lim_{x\to\infty} p(x) = \infty$
$\lim_{x\to-\infty} p(x) = \infty$

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