Question

find (a) the leading term of the polynomial, (b) the limit as x approaches ∞, and (c) the limit as x approaches -∞. p(x)=9x^2 + 6x^4+8x (a) the leading term of p(x) is

Explanation:

Step1: Identify leading - term

The leading term of a polynomial is the term with the highest degree. For the polynomial $p(x)=9x^{2}+6x^{4}+8x$, the degrees of the terms are 2, 4, and 1 respectively. The term with the highest degree is $6x^{4}$.

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

As $x\to\infty$, the behavior of the polynomial is dominated by the leading - term. For $y = 6x^{4}$, when $x\to\infty$, $y\to\infty$ since the coefficient 6 is positive and the exponent 4 is even. Mathematically, $\lim_{x\to\infty}(6x^{4}+9x^{2}+8x)=\lim_{x\to\infty}6x^{4}=\infty$.

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

As $x\to-\infty$, for the function $y = 6x^{4}$, since the exponent 4 is even, when we substitute a large negative number for $x$, $x^{4}$ is positive. And with a positive coefficient 6, $y\to\infty$. Mathematically, $\lim_{x\to-\infty}(6x^{4}+9x^{2}+8x)=\lim_{x\to-\infty}6x^{4}=\infty$.

Answer:

(A) $6x^{4}$
(B) $\infty$
(C) $\infty$