Final answer:
To find vertical asymptotes of a rational function, set the denominator to zero and solve for the variable; for horizontal asymptotes, compare degrees of the numerator and denominator polynomials, and find the asymptote accordingly.
Step-by-step explanation:
To find the vertical and horizontal asymptotes of a rational function, follow these steps:
Start by putting the rational function in its simplest form, where the numerator and denominator are both polynomials.
For vertical asymptotes, set the denominator equal to zero and solve for the variable. These are the values that the function cannot take, hence creating a vertical asymptote where the function approaches infinity.
For horizontal asymptotes, compare the degrees of the polynomials in the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0. If the degrees are equal, divide the leading coefficients, this ratio is the horizontal asymptote. If the numerator's degree is greater than the denominator's, there is no horizontal asymptote.
An example of a function with asymptotes is y = 1/x. Here, there is a vertical asymptote at x=0 and a horizontal asymptote at y=0, since as x approaches zero, y approaches infinity, and vice versa.