Question

Let f: R^2 -> R be defined by f(x, y) = (x²y) / (x^4 + y²) (x, y) ≠ (0,0). Show that

lim (x, y) -> (0,0) f(x, y) does not exist

Answer 1

To show that the limit lim (x, y) -> (0,0) f(x, y) does not exist, we can find two different paths that approach (0,0) and give different values for f(x, y). The paths x = 0 and y = 0 both give a limit of 0, but the path y = x² gives a limit of 1/2. Therefore, the limit does not exist.

Step-by-step explanation:

To show that the limit lim (x, y) -> (0,0) f(x, y) does not exist, we need to find two different paths that approach (0,0) and give different values for f(x, y).

Let's consider the paths x = 0 and y = 0. When x = 0, the limit becomes lim (0, y) -> (0,0) f(0, y) = (0²y) / (0^4 + y²) = 0. And when y = 0, the limit becomes lim (x, 0) -> (0,0) f(x, 0) = (x²0) / (x^4 + 0²) = 0.

Since the two paths give the same value of 0, lim (x, y) -> (0,0) f(x, y) appears to exist. However, if we consider the path y = x², the limit becomes lim (x, x²) -> (0,0) f(x, x²) = (x²(x²)) / (x^4 + (x²)²) = x^4 / (x^4 + x^4) = 1/2. Therefore, the limit does not exist as the value depends on the path.