Home > APCALC > Chapter 1 > Lesson 1.4.4 > Problem1-186

1-186.

Write an equation for the end-behavior function of $f(x)=\frac{x^3+3x^2-4x-1}{x^2-1}$. Then, write a complete set of approach statements for $f$.

This can be found using polynomial division.

$\frac{x^{3}+3x^{2}-4x-1}{x^{2}-1}=x+3+\frac{-3x+2}{x^{2}-1}$

Make sure to show your steps.

$b\left(x\right) = x + 3$

$\text{The term }\frac{-3x+2}{x^{2}-1}\text{ is insignificant for a large positive and negative }x.$

How does the graph behave as $x$ approaches $∞, −∞$, as well as the asymptotes $1$ and $−1$ from both the positive and negative sides.