Final answer:
The correct answer is option A. The asymptotes of the tangent function are vertical and located at odd multiples of π/2, corresponding to the positions where the cosine function equals zero and therefore the tangent function is undefined.
Step-by-step explanation:
Asymptotes act as boundaries that a graph approaches but never crosses or reaches. For the tangent function, which is periodic and defined as the ratio of the sine and cosine functions, we have vertical asymptotes where the cosine function is equal to zero since division by zero is undefined.
The cosine function equals zero at odd multiples of π/2. Therefore, vertical asymptotes for the tangent function occur at these points. To be more specific, for the function y = tan(x), vertical asymptotes are located at x = (2n+1)π/2 for every integer value of n. These asymptotes occur because, at these points, the cosine component of the tangent function (since tan(x) = sin(x)/cos(x)) equals zero, leading to an undefined value for the tangent.
Contrary to vertical asymptotes, the tangent function does not have horizontal asymptotes because its values continue to increase or decrease without bound as it approaches the vertical asymptotes.