Question

find (a) the leading term of the polynomial, (b) the limit as x approaches ∞, and (c) the limit as x approaches -∞. p(x) = 8x⁶ - 2x⁸ + 6x (a) the leading term of p(x) is \boxed{}. (b) the limit of p(x) as x approaches ∞ is \boxed{}. (c) the limit of p(x) as x approaches -∞ is \boxed{}.

Explanation:

Step1: Find the leading term

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

Step2: Find \(\lim_{x

ightarrow\infty}p(x)\)
As \(x
ightarrow\infty\), the leading term \(-2x^{8}\) dominates.
\(\lim_{x
ightarrow\infty}p(x)=\lim_{x
ightarrow\infty}(-2x^{8}+8x^{6}+6x)\)
Since \(\lim_{x
ightarrow\infty}ax^{n}=-\infty\) when \(a < 0\) and \(n\) is even (\(n = 8\), \(a=-2\))
\(\lim_{x
ightarrow\infty}p(x)=-\infty\)

Step3: Find \(\lim_{x

ightarrow-\infty}p(x)\)
As \(x
ightarrow-\infty\), the leading term \(-2x^{8}\) dominates.
\(\lim_{x
ightarrow-\infty}p(x)=\lim_{x
ightarrow-\infty}(-2x^{8}+8x^{6}+6x)\)
Since \(\lim_{x
ightarrow-\infty}ax^{n}=-\infty\) when \(a < 0\) and \(n\) is even (\(n = 8\), \(a = - 2\))
\(\lim_{x
ightarrow-\infty}p(x)=-\infty\)

Answer:

(A) \(-2x^{8}\)
(B) \(-\infty\)
(C) \(-\infty\)