Final answer:
The limit of the function f(x,y) = xy/(x²+y²) does not exist as (x,y) approaches (0,0) because different paths towards the origin result in different limit values.
Step-by-step explanation:
We are asked to show that the limit does not exist for the function f(x,y) = xy/(x²+y²) as (x,y) approaches (0,0). To demonstrate this, we can approach the origin along different paths and show that the limit varies depending on the path taken, hence the limit at the point does not exist.
For example, suppose we take the path y = x, then the function simplifies to:
f(x,x) = βx/(x²+x²) = x²/(2x²) = 1/2
As (x,y) → (0,0) along y=x, the limit appears to be 1/2. However, if we take the path y = mx (where m is a constant), the function becomes:
f(x,mx) = mx²/(x²+m²x²) = m/(1+m²)
In this case, the limit value depends on the slope m. Since the limit can have multiple values depending on the path taken to approach (0,0), it does not exist according to the definition of limits in multivariable calculus.