Final Answer:
The limit as (x, y) approaches (1, 29) of x/√(x+y) is undefined. The function is continuous everywhere except at the point (0, 0).
Step-by-step explanation:
The given function is f(x, y) = x/√(x+y). To find the limit as (x, y) approaches (1, 29), substitute these values into the function:
lim (x,y)→(1,29) x/√(x+y)
= 1/√(1+29)
= 1/√30
Since the denominator is nonzero, the limit exists and is equal to 1/√30. However, the limit is undefined at (0, 0) as the denominator becomes zero, indicating a potential discontinuity at this point.
Now, let's discuss the continuity of the function. The function is continuous everywhere except at points where the denominator becomes zero, which is x+y=0. This implies that the function is continuous for x+y≠0.
Therefore, the function is continuous everywhere except at the point (0, 0). In other words, the function is continuous for x+y>0, making it undefined for x+y=0 and continuous for x+y<0. This analysis is crucial for understanding the behavior of the function across different regions of the xy-plane.