Asymptotes are lines that a graph approaches but never actually touches or crosses at infinity. There are three main types of asymptotes:

1. Horizontal Asymptote

• A horizontal line that the graph approaches as $x$ tends towards infinity or negative infinity.
• Typically found in rational functions where the degree of the polynomial in the numerator is less than or equal to that of the denominator.
• Example: For $f(x) = \frac{1}{x}$, the horizontal asymptote is $y = 0$ as $x \to \infty$ or $x \to -\infty$.

2. Vertical Asymptote

• A vertical line that the graph approaches but never crosses, usually where the function is undefined.
• These occur when a denominator of a rational function equals zero (and the numerator is non-zero).
• Example: For $f(x) = \frac{1}{x-2}$, the vertical asymptote is at $x = 2$, because the function is undefined at this point.

3. Oblique (Slant) Asymptote

• A slanted line that the graph approaches as $x \to \infty$ or $x \to -\infty$.
• These appear when the degree of the numerator is exactly one more than the degree of the denominator in a rational function.
• Example: For $f(x) = \frac{x^2 + 1}{x}$, the oblique asymptote is $y = x$, found by performing polynomial division.

Special Note: Curved Asymptotes

• Some graphs may approach a curved asymptote, but these are less common and not typically included in elementary discussions of asymptotes.

Would you like more details or have any questions?

5 Related Questions:

1. How do you find the horizontal asymptote of a rational function?
2. What is the difference between a vertical asymptote and a hole in a graph?
3. How can you determine if a function has an oblique asymptote?
4. Can a graph have more than one horizontal asymptote?
5. What happens to the behavior of a function near a vertical asymptote?

Tip: Always check the degrees of the numerator and denominator when dealing with rational functions to quickly assess if there are horizontal or oblique asymptotes.