Find (A) the leading term of the polynomial, (B) the limit as $$ x $$ approaches $$ \infty $$, and (C) the limit as $$ x $$ approaches $$ -\infty $$. $$ p(x)=18+2 x^{6}-5 x^{7} $$ (A) The leading term is $$ -5 x^{7} $$. (B) The limit of $$ p(x) $$ as $$ x $$ approaches $$ \infty $$ is $$ \square $$.

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Certainly! Let's analyze the polynomial $$ p(x) = 18 + 2x^{6} - 5x^{7} $$ step by step.

### (A) Leading Term
The **leading term** of a polynomial is the term with the highest degree (the largest exponent of $$ x $$).

For $$ p(x) = 18 + 2x^{6} - 5x^{7} $$:
- The degrees of the terms are:
  - $$ 18 $$ has degree 0.
  - $$ 2x^{6} $$ has degree 6.
  - $$ -5x^{7} $$ has degree 7.

**Leading Term:** $$ -5x^{7} $$

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### (B) Limit as $$ x $$ Approaches $$ \infty $$

To find $$ \lim_{x \to \infty} p(x) $$, we consider the behavior of the leading term since it dominates the polynomial for large $$ x $$.

- **Leading Term:** $$ -5x^{7} $$

As $$ x \to \infty $$:
- $$ x^{7} $$ grows without bound towards $$ +\infty $$.
- Multiplying by $$ -5 $$, the term $$ -5x^{7} $$ tends to $$ -\infty $$.

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

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### (C) Limit as $$ x $$ Approaches $$ -\infty $$

Similarly, to find $$ \lim_{x \to -\infty} p(x) $$, we analyze the leading term.

- **Leading Term:** $$ -5x^{7} $$

As $$ x \to -\infty $$:
- For odd exponents, $$ x^{7} $$ behaves as $$ x $$ does. So, $$ x^{7} $$ tends to $$ -\infty $$.
- Multiplying by $$ -5 $$, the term $$ -5x^{7} $$ tends to $$ +\infty $$.

**Limit:** 
$$
\lim_{x \to -\infty} p(x) = +\infty
$$

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### Summary

- **(A)** The leading term is $$ -5x^{7} $$.
- **(B)** $$ \lim_{x \to \infty} p(x) = -\infty $$.
- **(C)** $$ \lim_{x \to -\infty} p(x) = +\infty $$.
(A) The leading term is $$ -5x^{7} $$. (B) The limit of $$ p(x) $$ as $$ x $$ approaches $$ \infty $$ is $$ -\infty $$. (C) The limit of $$ p(x) $$ as $$ x $$ approaches $$ -\infty $$ is $$ +\infty $$.

Find (A) the leading term of the polynomial, (B) the limit as \( x \) approaches \( \infty \), and (C) the limit as \( x \) approaches \( -\infty \). \( p(x)=18+2 x^{6}-5 x^{7} \) (A) The leading term is \( -5 x^{7} \). (B) The limit of \( p(x) \) as \( x \) approaches \( \infty \) is \( \square \).

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