Question

question 12 - 1 point for the function f(x) given below, evaluate lim x→∞ f(x) and lim x→-∞ f(x). f(x) = (x³ + 3)/(-x³ + 3x²) provide your answer below. lim x→∞ f(x) = □ lim x→-∞ f(x) = □

Explanation:

Step1: Divide numerator and denominator by highest - power of x

For $\lim_{x
ightarrow\infty}f(x)$ and $\lim_{x
ightarrow-\infty}f(x)$ where $f(x)=\frac{x^{3}+3}{-x^{3}+3x^{2}}$, divide both the numerator and denominator by $x^{3}$. We get $f(x)=\frac{1 + \frac{3}{x^{3}}}{-1+\frac{3}{x}}$.

Step2: Evaluate $\lim_{x

ightarrow\infty}f(x)$
As $x
ightarrow\infty$, $\frac{3}{x^{3}}
ightarrow0$ and $\frac{3}{x}
ightarrow0$. So, $\lim_{x
ightarrow\infty}\frac{1+\frac{3}{x^{3}}}{-1 + \frac{3}{x}}=\frac{1 + 0}{-1+0}=- 1$.

Step3: Evaluate $\lim_{x

ightarrow-\infty}f(x)$
As $x
ightarrow-\infty$, $\frac{3}{x^{3}}
ightarrow0$ and $\frac{3}{x}
ightarrow0$. So, $\lim_{x
ightarrow-\infty}\frac{1+\frac{3}{x^{3}}}{-1+\frac{3}{x}}=\frac{1 + 0}{-1+0}=-1$.

Answer:

$\lim_{x
ightarrow\infty}f(x)=-1$
$\lim_{x
ightarrow-\infty}f(x)=-1$