Final answer:
Asymptotes in rational functions are determined by comparing the degrees of the numerator and rationalized denominator.
Step-by-step explanation:
When dealing with rational functions, which are ratios of polynomials, various types of asymptotes can be determined based on the degrees of the numerator and denominator when the denominator has a square root.
If the degree of the numerator is less than the degree of the polynomial in the denominator, after rationalizing, the horizontal asymptote is y = 0. When degrees are equal, the horizontal asymptote is found by the ratio of the leading coefficients
A vertical asymptote occurs at values of x where the denominator equals zero, causing the function to approach infinity. To find this, solve the equation set by the denominator equal to zero, except for when the same factor is canceled out by the numerator.
An oblique or slant asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator after rationalization. To find this asymptote, perform polynomial long division or synthetic division.
Factors with square roots in the denominator should be simplified, possibly by multiplying by the conjugate to rationalize and then examining the resulting polynomial's degrees to find the asymptotes.