In Exercises 21–36, find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x^2+4)

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function r(x) = x/(x^2 + 4), the numerator is a polynomial of degree 1, and the denominator is a polynomial of degree 2. Understanding the structure of rational functions is essential for analyzing their behavior, including identifying asymptotes and holes.

Vertical asymptotes occur in a rational function when the denominator approaches zero while the numerator does not simultaneously approach zero. To find vertical asymptotes, we set the denominator equal to zero and solve for x. In the function r(x), we need to determine where x^2 + 4 = 0, which helps identify the locations of any vertical asymptotes.

Holes in the graph of a rational function occur at values of x where both the numerator and denominator equal zero, indicating a removable discontinuity. To find holes, we factor both the numerator and denominator and identify common factors. In the function r(x), we check if there are any common factors that would lead to a hole in the graph.