1. **State the problem:** We need to identify and list the holes, vertical asymptotes, and horizontal asymptotes of a rational function. A rational function is a ratio of two polynomials, typically written as $$f(x) = \frac{P(x)}{Q(x)}$$ where $P(x)$ and $Q(x)$ are polynomials. 2. **Important rules:** - **Holes** occur where both numerator and denominator are zero at the same $x$ value, meaning a common factor cancels out. - **Vertical asymptotes** occur where the denominator is zero but the numerator is not zero at that $x$ value. - **Horizontal asymptotes** describe the behavior of $f(x)$ as $x \to \pm \infty$ and depend on the degrees of $P(x)$ and $Q(x)$: - If degree of $P < $ degree of $Q$, horizontal asymptote is $y=0$. - If degree of $P = $ degree of $Q$, horizontal asymptote is $y=\frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q}$. - If degree of $P > $ degree of $Q$, no horizontal asymptote (there may be an oblique asymptote). 3. **Steps to identify:** - Factor numerator $P(x)$ and denominator $Q(x)$ completely. - Find common factors between numerator and denominator; the zeros of these common factors are holes. - Find zeros of denominator after canceling common factors; these are vertical asymptotes. - Determine degrees of numerator and denominator to find horizontal asymptotes using the rules above. 4. **Example:** Suppose $$f(x) = \frac{(x-2)(x+3)}{(x-2)(x-5)}$$ - Common factor: $(x-2)$, so there is a hole at $x=2$. - After canceling $(x-2)$, denominator zero at $x=5$ gives vertical asymptote at $x=5$. - Degrees of numerator and denominator after cancellation are both 1, so horizontal asymptote is $y=\frac{1}{1}=1$. 5. **Summary:** - Holes: points where factors cancel. - Vertical asymptotes: zeros of denominator after cancellation. - Horizontal asymptotes: determined by degrees of numerator and denominator. This method applies to any rational function to identify holes and asymptotes.