question\ndetermine if the function below has a horizontal or a slant asymptote, and explain why.\n…

question\ndetermine if the function below has a horizontal or a slant asymptote, and explain why.\n f(x)=\frac{-15x^{2}+17 + 2x+5x^{3}}{-4x + x^{2}+4} \nanswer attempt 1 out of 2\nthe function has a \nasymptote because the degree of the numerator is \nthe degree of the denominator.\nas ( x ) approaches infinity (positive or negative), the value of ( y ) approaches.

question\ndetermine if the function below has a horizontal or a slant asymptote, and explain why.\n f(x)=\frac{-15x^{2}+17 + 2x+5x^{3}}{-4x + x^{2}+4} \nanswer attempt 1 out of 2\nthe function has a \nasymptote because the degree of the numerator is \nthe degree of the denominator.\nas ( x ) approaches infinity (positive or negative), the value of ( y ) approaches.

Answer

Explanation:

Step1: Identify degrees of numerator and denominator

The numerator $-15x^{2}+17 + 2x+5x^{3}$ has degree 3 (highest - power of $x$ is 3), and the denominator $-4x + x^{2}+4$ has degree 2 (highest - power of $x$ is 2).

Step2: Determine type of asymptote

Since the degree of the numerator (3) is greater than the degree of the denominator (2), the function has a slant asymptote.

Step3: Find the slant asymptote

Use polynomial long - division: Divide $5x^{3}-15x^{2}+2x + 17$ by $x^{2}-4x + 4$. [ \begin{align*} \frac{5x^{3}-15x^{2}+2x + 17}{x^{2}-4x + 4}&=5x + 5+\frac{12x - 3}{x^{2}-4x + 4} \end{align*} ] As $x\to\pm\infty$, $\frac{12x - 3}{x^{2}-4x + 4}\to0$. So as $x$ approaches infinity (positive or negative), the value of $y$ approaches $5x + 5$.

Answer:

The function has a slant asymptote because the degree of the numerator is greater than the degree of the denominator. As $x$ approaches infinity (positive or negative), the value of $y$ approaches $5x + 5$.