Find (A) the leading term of the polynomial, (B) the limit as approaches , and © the limit as approaches .

(A) The leading term is .
(B) The limit of as approaches is .

Ask by Cook Adkins. in the United States

Jan 22,2025

Question

Find (A) the leading term of the polynomial, (B) the limit as approaches , and © the limit as approaches .

(A) The leading term is .
(B) The limit of as approaches is .

Ask by Cook Adkins. in the United States

Jan 22,2025

UpStudy AI · Accepted Answer

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

---

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

---

### 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 approaches , and © the limit as approaches .

(A) The leading term is .
(B) The limit of as approaches is .

Solución de inteligencia artificial de Upstudy

Responder

Solución

(A) Leading Term

(B) Limit as Approaches

© Limit as Approaches

Summary

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions

Find (A) the leading term of the polynomial, (B) the limit as approaches , and © the limit as approaches . (A) The leading term is . (B) The limit of as approaches is .

Solución de inteligencia artificial de Upstudy

Responder

Solución

(A) Leading Term

(B) Limit as Approaches

© Limit as Approaches

Summary

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions

Find (A) the leading term of the polynomial, (B) the limit as approaches , and © the limit as approaches .

(A) The leading term is .
(B) The limit of as approaches is .