Showing that a limit does not exist

Question

lim(x,y)→(1,0) [(y² + x - 1)²] / [(x - 1)² + y⁴)]

Answer 1

Let (x, y)→(1, 0) along line x = y² + 1 Then lim (x,y)→(1, 0) [(y² + x - 1)²] / [(x - 1)² + y⁴)] = lim y→0

(2y²)²/ (2y⁴) = 2. Let (x,y)→(1,0) along line x = 2y² + 1. Then [(y² + x - 1)²] / [(x - 1)² + y⁴)] = (3y²)²/(5y⁴)→9/5.

So, given limit DNE