Final answer:
The limit lim(x,y)→(0,0) (4xy⁴) / (x² + y⁸) exists and is equal to 0.
Step-by-step explanation:
To find the limit lim(x,y)→(0,0) (4xy⁴) / (x² + y⁸), we need to approach (0,0) along different paths and see if the limit exists and if it is the same for all paths.
First, let's approach (0,0) along the x-axis by setting y = 0:
lim(x,0)→(0,0) (4x(0)⁴) / (x² + (0)⁸) = 0 / x² = 0
Since the limit is 0 when approaching along the x-axis, let's now approach along the y-axis by setting x = 0:
lim(0,y)→(0,0) (4(0)y⁴) / ((0)⁰ + y⁸) = 0 / y⁸ = 0
Again, the limit is 0 when approaching along the y-axis. Since the limit is the same for both paths, we can conclude that the limit lim(x,y)→(0,0) (4xy⁴) / (x² + y⁸) exists and is equal to 0.