Final Answer:
Horizontal Asymptote: y = 0. Vertical Asymptote: x = -2
Step-by-step explanation:
For rational functions like r(x) = 3/(x+2), the presence of horizontal and vertical asymptotes can be determined based on the behavior of the function as x approaches certain values.
Horizontal asymptotes occur when the function approaches a constant value as x moves toward positive or negative infinity. In this case, the degree of the denominator (x + 2) is higher than the degree of the numerator (3), resulting in a horizontal asymptote at y = 0 (the x-axis). As x becomes extremely large (positive or negative), the function values approach but never quite reach zero, establishing the horizontal asymptote.
Vertical asymptotes arise where the function's denominator becomes zero, causing the function to be undefined at those points. In the function r(x) = 3/(x+2), the denominator x + 2 becomes zero when x = -2. Therefore, x = -2 is a vertical asymptote. As x approaches -2 from either side, the function's values tend to infinity or negative infinity, indicating the presence of a vertical asymptote at x = -2.
Hence, the function r(x) = 3/(x+2) has a horizontal asymptote at y = 0 and a vertical asymptote at x = -2, delineating the behavior of the function as x moves towards positive or negative infinity and its discontinuity at x = -2.