Final answer:
The limit of the function e^y² tan(xz) as it approaches the point (π, 0, 1/3) exists and the function is continuous there, but the correct limit value is not included in the given options; it is actually tan(π/3), which is approximately √3.
Step-by-step explanation:
The student is asking whether the limit of the function ey²tan(xz) as it approaches the point (π, 0, 1/3) is continuous and what the value of the limit is. To determine the continuity and the limit, we need to consider the behavior of the exponential function and the tangent function separately. The exponential function ey² is continuous everywhere, and as y approaches 0, ey² approaches 1. The tangent function, however, is discontinuous at x = (2n+1)π/2 for integers n. Since we are considering x = π and z = 1/3, tan(xz) = tan(π/3) which is defined and ≈ √3. This means that the function ey²tan(xz) is continuous at the point in question, and the limit as y approaches 0 would be tan(π/3) ≈ √3, not 0, 1, or π. Therefore, the given options do not include the correct answer, and the response would be that the limit exists but the correct limit is not listed in the options.