Multivariable functions limits and paths In order to approach a point as (0,0) there many directions to do so.
A whole 360 degrees actually.
So between [0,360) degrees there are actually infinite directions.
My question is why does it matter for the value of a limit of say a 2 variable function if the path to that point is linear or non linear.
For example from wikipedia article on multivariable calculus on limit section 
$$ 
f(x,y) = \frac{x^2y}{x^4 + y^2} 
$$
it says when approaching to (0,0) from any line the limit is the same,
but if we do it along a parabola the result is different.
Ultimately in both cases the point is approached from a particular degree.
(Although I am not really sure what happens on the parabola case or if we can even say this for the parabola's case or any non linear path)
Why when the degree of approachment is same both for the parabola and line,the limit is different.
Or have I got it all wrong assuming that a non linear path can finally approach a point from a particular direction?
 A: Here's what might be an interesting counter example to your assumption that paths have to approach in a way that is similar to some particular linear path. Consider if we are taking a limit of some function $f(x,y)$ as $(x,y)$ approaches $(0,0)$. Then all linear paths would be $x=0$ and $y=mx$ where $m$ is any real number.
A different path that approaches $(0,0)$ is $x=\frac{\cos \left(t\right)}{t},\:y=\frac{\sin \left(t\right)}{t}$
Spiral towards origin
This path approaches the origin, but never even has a tangent line that is like any of the linear paths mentioned earlier. In fact, it seems that the tangent line of this curve as $t$ increases becomes closer and closer to perpendicular to any of the previous linear paths that happen to pass through the same point on this new curve.
A: If the limit does exist, you can choose any path, and the result will be the same. On the other hand, you can check ten thousand paths that will give you the same result, but you cannot guarantee that the limit exists. It could always exist that one messy path that spoils it all. But once you got a result using a path, of the limit is to exist, it will be that number. Using paths is good only for proving the non-existance of a limit. 
The conclusion is straightforward forward: if using two paths (e.g. a line and a parabola), you get different results, the limit does not exist. Don't fret over it too much. 
Suppose that $(x,y) \to {\bf 0}$. You can use a lot more thing that just lines and parabolas, for example:


*

*polynomials of any degree (lines and parabolas included): $(t, a_1t + \cdot + a_nt^n)$ with $t\to 0 $ (take whatever $n \in \Bbb Z_{>0}$ you want). If the result depends on any of the constants $a_j$, the limit does not exist.

*polar coordinates: $(r\cos \theta, r\sin\theta $ with $r\to 0$. If the result depends on $\theta $, the limit does not exists.

*waves: $(t, a_1\sin t + a_2 \sin(2t) + \cdots + a_n\sin(nt))$, with $t\to 0 $. Same conclusion as the first example.

*in general, take any $f:\Bbb R\to \Bbb R$, continuous, for our comfort, such that $f(0) = 0$, and take $(t, f(t))$ with $t\to 0$. Choose the $f$ which is better for your particular problem. 

*yet more general, $(f(t), g(t))$, in the same catch as above. But usually the first examples above will do.
My point is: there is nothing particular about lines and parabolas, and degree of approachment. Think outside the box :)
