Questions: Find all asymptotes and/or holes in the graph of (f(x)=fracx^2-2 x-8x^2-x-6).

Solution Steps

To find the asymptotes and holes in the graph of the function $f(x) = \frac{x^2 - 2x - 8}{x^2 - x - 6}$, we need to:

1. Factor both the numerator and the denominator.
2. Identify any common factors between the numerator and the denominator, which will indicate holes.
3. Determine the vertical asymptotes by setting the denominator equal to zero and solving for $x$.
4. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator.

The function $f(x) = \frac{x^2 - 2x - 8}{x^2 - x - 6}$ can be factored as follows:

• Numerator: $x^2 - 2x - 8 = (x - 4)(x + 2)$
• Denominator: $x^2 - x - 6 = (x - 3)(x + 2)$

The common factor between the numerator and the denominator is $x + 2$. This indicates that there is a hole in the graph at $x = -2$.

To find the vertical asymptotes, we set the denominator equal to zero: $$x - 3 = 0 \implies x = 3$$ Thus, there is a vertical asymptote at $x = 3$.

Since the degrees of the numerator and denominator are both 2, we find the horizontal asymptote by taking the limit as $x$ approaches infinity: $$\text{Horizontal Asymptote} = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{1}{1} = 1$$

Final Answer