Question

determine whether the limit is equal to ∞ or - ∞.
lim - 2x³ + 9x²
x→ - ∞
the limit is equal to because the degree of the polynomial is and the leading coefficient is

Explanation:

Step1: Analyze the degree of the polynomial

For the polynomial \( - 2x^{3}+9x^{2}\), the degree \(n = 3\) (the highest power of \(x\)).

Step2: Analyze the leading coefficient

The leading term is \( - 2x^{3}\), and the leading coefficient \(a=-2\).

Step3: Determine the limit as \(x\to-\infty\)

For a polynomial \(P(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_0\) with \(n\) odd (\(n = 3\)) and \(a_n=-2<0\).
When \(x\to-\infty\), and \(n\) is odd, \(P(x)\to+\infty\) if \(a_n>0\) and \(P(x)\to-\infty\) if \(a_n < 0\).

Answer:

The limit is equal to \(-\infty\) because the degree of the polynomial is \(3\) (odd) and the leading coefficient is \(- 2\) (negative).