Question

find (a) the leading term of the polynomial, (b) the limit as x approaches ∞, and (c) the limit p(x)=2x + x^5 - 9x^4 (a) the leading term is . (b) the limit of p(x) as x approaches ∞ is . (c) the limit of p(x) as x approaches -∞ is .

Explanation:

Step1: Identify leading term

The leading term of a polynomial is the term with the highest - degree. For \(p(x)=2x + x^{5}-9x^{4}\), the degrees of the terms \(2x\), \(x^{5}\), and \(-9x^{4}\) are 1, 5, and 4 respectively. So the leading term is \(x^{5}\).

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

As \(x\to\infty\), the behavior of the polynomial \(p(x)\) is determined by its leading term. Since the leading term is \(x^{5}\) (and the coefficient of \(x^{5}\) is 1>0), \(\lim_{x\to\infty}(2x + x^{5}-9x^{4})=\lim_{x\to\infty}x^{5}=\infty\).

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

For a polynomial \(y = a_nx^n+\cdots+a_0\) with \(n = 5\) (odd) and \(a_n = 1>0\), \(\lim_{x\to-\infty}x^{5}=-\infty\). So \(\lim_{x\to-\infty}(2x + x^{5}-9x^{4})=\lim_{x\to-\infty}x^{5}=-\infty\).

Answer:

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