I'm trying to review for my final and need help solving this problem that was given in a past homework assignment. The answer I keep getting is $-4/3$, but the correct answer is the limit DNE. I know I can approach this problem by testing different paths, but the way my professor taught it was to use the math $y = mx$. Can someone please guide me through this step-by-step?

$$\lim_{(x, y) \to (0, 0)} \frac{\sin(2x) - 2x + y}{x^3 + y}$$