limit of $\frac{2xy^3}{7x^2+4y^6}$, different answers. Good evening, everyone I've tried my possible best to evaluate the limit as $(x,y) \to (0,0)$ but Using sage the answer is 0 either ways but one textbook is saying the limit doesn't exist.  Could the 2 answers be correct?

My guess is the problem is that sage cannot test every approach curve, so it probably tests a small grid of points about the origin. Since the place where the limit is bad is shaoed like a curved cubic, the grid will miss the curve and give the wrong answer.
Note that Wolfram Alpha has the same problem: http://m.wolframalpha.com/input/?i=limit+as+x+goes+to++0%2C+y+goes+to+0+of+%5Cfrac%7B2xy%5E3%7D%7B7x%5E2%2B4y%5E6%7D&incCompTime=true
