Final answer:
The limit of f(x,y) as (x,y) approaches (0,0) does not exist along the path x=0, is zero along the path y=0, and is zero along the path y=3x.
Step-by-step explanation:
To evaluate the limit of the function f(x,y) = x²+y² / (xy+y³) as (x,y) approaches (0,0) along the specified paths, we can substitute the path equations into the function and simplify to find the limits.
Along the path x=0 :
Substituting x = 0 into the function, we get f(0,y) which is undefined because the function will have y² in the numerator and a zero multiply y in the denominator, hence the limit does not exist (DNE).
Along the path y=0 :
Substituting y = 0 into the function, we get f(x,0) which is zero because the numerator and denominator will be zero, and zero divided by zero can be defined as zero for this evaluation as x approaches 0.
Along the path y=3x :
Substituting y = 3x into the function, we simplify to find that as x approaches 0, the limit also approaches zero because the terms containing x in the numerator and denominator cancel out.