This image seems to contain examples and explanations about vertical and horizontal asymptotes of functions. There are also some mathematical operations and clarifications related to these concepts.

Here's a breakdown of the content:

1. Vertical Asymptotes (A.V.): These occur where the denominator of a rational function equals zero, provided that the numerator doesn't also become zero at the same point. For example:

   • $f(x) = \frac{1}{x-2}$ implies $x = 2$ is a vertical asymptote.

2. Horizontal Asymptotes (A.H.): These relate to the behavior of the function as $x$ tends to infinity. It depends on the degrees of the polynomials in the numerator and denominator:

   • If the degree of the numerator is less than the denominator, the horizontal asymptote is at $y = 0$.
   • If they are equal, the horizontal asymptote is at $y = \frac{a_n}{b_m}$, where $a_n$ and $b_m$ are the leading coefficients of the numerator and denominator.

3. Examples of finding vertical and horizontal asymptotes:

   • Examples like $f(x) = \frac{x^3 + 3}{x^2 - 16}$ and $f(x) = \frac{-5x}{x(x+1)}$ show how to find vertical and horizontal asymptotes.
   • The page also mentions cases where no vertical or horizontal asymptotes exist.

If you would like a deeper explanation of specific examples or need help solving some related problems, feel free to ask!

Would you like more details on a specific example or concept? Or do you have any questions about the content?

Here are 5 related questions:

1. How do you find the vertical asymptotes of rational functions?
2. Can horizontal asymptotes exist in functions that are not rational?
3. How do you determine the degrees of polynomials in rational functions?
4. What happens when the numerator and denominator have the same degree?
5. How can we graph a function that has both vertical and horizontal asymptotes?

Tip: Always check the degrees of the numerator and denominator when looking for horizontal asymptotes!