Question

find (a) the leading term of the polynomial, (b) the limit as x approaches ∞, and (c) the limit as x approaches - ∞. p(x)=4x + x^7 - 8x^6 (a) the leading term is □.

Explanation:

Step1: Identify leading term

The leading term of a polynomial is the term with the highest degree. In $p(x)=4x + x^{7}-8x^{6}$, the term with the highest degree is $x^{7}$.

Step2: Find limit as $x\to\infty$

As $x\to\infty$, for a polynomial $a_nx^n+\cdots+a_1x + a_0$ with $a_n
eq0$, the limit is determined by the leading - term. Here $n = 7$ (odd) and the leading - coefficient of $x^{7}$ is $1>0$. So $\lim_{x\to\infty}(4x + x^{7}-8x^{6})=\lim_{x\to\infty}x^{7}=\infty$.

Step3: Find limit as $x\to-\infty$

Since the degree $n = 7$ (odd) and the leading - coefficient of $x^{7}$ is $1>0$, $\lim_{x\to-\infty}(4x + x^{7}-8x^{6})=\lim_{x\to-\infty}x^{7}=-\infty$.

Answer:

(A) $x^{7}$
(B) $\infty$
(C) $-\infty$