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^4 + 8x + 5x^6. (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 \(p(x)=9x^{4}+8x + 5x^{6}\), the degrees of the terms are 4, 1, and 6 respectively. The term with the highest - degree is \(5x^{6}\).

Step2: Find limit as \(x\to\infty\)

As \(x\to\infty\), for a polynomial \(y = a_nx^n+\cdots+a_0\) (\(a_n
eq0\)), \(\lim_{x\to\infty}p(x)=\lim_{x\to\infty}a_nx^n\) (the leading - term dominates). Here \(a_n = 5\) and \(n = 6\). Since \(n = 6\) (even) and \(a_n=5>0\), \(\lim_{x\to\infty}(5x^{6})=\infty\).

Step3: Find limit as \(x\to-\infty\)

As \(x\to-\infty\), for the leading term \(y = 5x^{6}\), when we substitute \(x=-t\) (\(t\to\infty\)), we have \(y = 5(-t)^{6}=5t^{6}\). Since \(n = 6\) (even) and \(a_n = 5>0\), \(\lim_{x\to-\infty}(5x^{6})=\infty\).

Answer:

(A) \(5x^{6}\)
(B) \(\infty\)
(C) \(\infty\)