Question

polynomial functions and end behavior
practice
describe the end behavior of each function using limit notation.

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

( limlimits_{x \to -infty} g(x) = -infty )
( limlimits_{x \to infty} g(x) = infty )

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) = -7x^9 - 8x^3 + 6 )

( limlimits_{x \to -infty} f(x) = infty )
( limlimits_{x \to infty} f(x) = -infty )

1. ( g(x) = -3x^6 + 5x^3 - 2x + 6 )

( limlimits_{x \to -infty} g(x) = -infty )
( limlimits_{x \to infty} g(x) = -infty )

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

( limlimits_{x \to -infty} f(x) = -infty )
( limlimits_{x \to infty} f(x) = infty )

1. ( p(x) = -8x^2 - 3x + 10 )

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

1. ( p(x) = -6x^5 + x^4 + 5x^2 - 4 )

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

1. ( f(x) = 3x^2 + 8x + 11 )

( limlimits_{x \to -infty} f(x) = infty )
( limlimits_{x \to infty} f(x) = infty )

1. (dots)
2. (dots)

Explanation:

Step1: Identify leading term for each polynomial

For a polynomial $a_nx^n+...+a_0$, focus on $a_nx^n$.

Step2: Apply end behavior rules

• If $n$ even: $\lim_{x\to\pm\infty}f(x)=\infty$ if $a_n>0$, $\lim_{x\to\pm\infty}f(x)=-\infty$ if $a_n<0$.
• If $n$ odd: $\lim_{x\to\infty}f(x)=\infty$ if $a_n>0$, $\lim_{x\to-\infty}f(x)=-\infty$; $\lim_{x\to\infty}f(x)=-\infty$ if $a_n<0$, $\lim_{x\to-\infty}f(x)=\infty$.

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

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

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

$\lim_{x\to\infty} g(x) = \infty$

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

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

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

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3. $f(x)=-7x^9-8x^3+6$

Leading term: $-7x^9$ (odd $n=9$, $a_n=-7<0$)

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

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4. $g(x)=-3x^6+5x^3-2x+6$

Leading term: $-3x^6$ (even $n=6$, $a_n=-3<0$)

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

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

Leading term: $x^3$ (odd $n=3$, $a_n=1>0$)

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

---

6. $p(x)=-8x^2-3x+10$

Leading term: $-8x^2$ (even $n=2$, $a_n=-8<0$)

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

---

7. $p(x)=-6x^5+x^4+5x^2-4$

Leading term: $-6x^5$ (odd $n=5$, $a_n=-6<0$)

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

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8. $f(x)=3x^2+8x+11$

Leading term: $3x^2$ (even $n=2$, $a_n=3>0$)

Step1: Evaluate $x\to-\infty$

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

Step2: Evaluate $x\to\infty$

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

Answer:

1. $\lim_{x\to-\infty} g(x) = -\infty$, $\lim_{x\to\infty} g(x) = \infty$
2. $\lim_{x\to-\infty} p(x) = \infty$, $\lim_{x\to\infty} p(x) = \infty$
3. $\lim_{x\to-\infty} f(x) = \infty$, $\lim_{x\to\infty} f(x) = -\infty$
4. $\lim_{x\to-\infty} g(x) = -\infty$, $\lim_{x\to\infty} g(x) = -\infty$
5. $\lim_{x\to-\infty} f(x) = -\infty$, $\lim_{x\to\infty} f(x) = \infty$
6. $\lim_{x\to-\infty} p(x) = -\infty$, $\lim_{x\to\infty} p(x) = -\infty$
7. $\lim_{x\to-\infty} p(x) = \infty$, $\lim_{x\to\infty} p(x) = -\infty$
8. $\lim_{x\to-\infty} f(x) = \infty$, $\lim_{x\to\infty} f(x) = \infty$