A function or curve can have asymptotes in one or both of the following ways:
- Vertical asymptotes: These occur when the function approaches infinity or negative infinity as the independent variable approaches a certain value. A function or curve can have vertical asymptotes at x = a if the function approaches infinity or negative infinity as x approaches a from the left or the right.
- Horizontal asymptotes: These occur when the function approaches a finite value as the independent variable goes to infinity or negative infinity. A function or curve can have a horizontal asymptote at y = b if the function approaches b as x goes to infinity or negative infinity.
Some common functions that have asymptotes include rational functions (functions that are fractions of polynomials), exponential functions, logarithmic functions, and trigonometric functions. For example, the rational function f(x) = (x^2 - 4) / (x - 2) has a vertical asymptote at x = 2, since the function approaches infinity as x approaches 2 from the left or the right. The exponential function f(x) = e^x has a horizontal asymptote at y = 0, since the function approaches 0 as x goes to negative infinity.
