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 \( \square \).

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:** \( -5x^{7} \) --- ### (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)** \( \lim_{x \to -\infty} p(x) = +\infty \).