Final answer:
A function has no asymptotes when it is continuous for all real numbers, doesn't approach a constant value at infinity, and lacks undefined points such as division by zero. Polynomial functions are typical examples of functions without asymptotes.
Step-by-step explanation:
A function may not have any asymptotes if it is well-behaved throughout its domain and does not approach infinity or have any kind of undefined points such as divisions by zero. Asymptotes are typically associated with functions that have singularities or certain behaviors that cause the function to tend towards infinity. There are two main types of asymptotes: vertical and horizontal. A vertical asymptote occurs when the values of the function grow arbitrarily large or small (approach infinity or negative infinity) as they get closer to some value of x. Horizontal asymptotes, on the other hand, occur when the function approaches a particular value as x approaches infinity or negative infinity.
For example, the function y = x^2 has no asymptotes because it is continuous for all real numbers and goes to infinity only as x itself goes to infinity. Furthermore, polynomial functions of any degree do not have asymptotes because they are continuous and do not have behavior that causes them to approach a specific value other than infinity. On the other hand, rational functions, like y = 1/x, may have both vertical and horizontal asymptotes; the function y = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote as x approaches infinity.
To summarize, there are no asymptotes for a function when the function is continuous and doesn't approach a constant value at infinity or has any type of undefined points like division by zero.