Question

describe the end behavior of the function f(x)=3x^2 + 4x^3+15 by finding lim f(x) and lim f(x) as x→∞ and x→ -∞. lim f(x) = (simplify your answer.) as x→∞

Explanation:

Step1: Identify the leading - term

The leading - term of the polynomial function \(f(x)=3x^{2}+4x^{3}+15\) is \(4x^{3}\) since the degree of \(4x^{3}\) (degree 3) is the highest among the terms.

Step2: Find \(\lim_{x

ightarrow\infty}f(x)\)
As \(x
ightarrow\infty\), we consider the behavior of the leading - term \(4x^{3}\). When \(x\) is very large and positive, \(4x^{3}\) is also very large and positive. So, \(\lim_{x
ightarrow\infty}(3x^{2}+4x^{3}+15)=\lim_{x
ightarrow\infty}4x^{3}=\infty\) because the leading - term dominates the behavior of the polynomial as \(x
ightarrow\infty\).

Answer:

\(\infty\)