How to compute this multi-variable limit? 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? 
$$\lim_{(x, y) \to (0, 0)} \frac{\sin(2x) - 2x + y}{x^3 + y}$$

You can do the Taylor expansion on $\sin 2x$ giving
$\frac {-\frac {8x^3}{6} + y}{x^3 + y}$
If we set y=0 and let x approach zero, the limit would appear to be $-\frac 43$
but if we set x = 0 and let y approach 0 the limit would appear to be $1$
$\lim_\limits{x\to 0} f(x,0) \ne \lim_\limits{y\to 0} f(0,y)$
If the limit exists, it must be the same along all paths toward 0.
