why can't we check for existence of limits by only checking along all the possible lines through the point at which the limit is evaluated? We generally check for limits of a multivariable function by checking limits at that point along various paths. But wouldn't it be just enough to check the limit of all possible lines passing through the point, as it seems to me as equivalent of approaching the point from all the directions? and also can't we approximate the curve of any other function to a line near the small neighborhood (around the point which the limit is being evaluated) and wouldn't it be equivalent to checking the limits along the tangent of the function (which would mean we have already checked it when we checked the limit of all possible lines passing through the point)?
But this doesn't seem to work,
for example

here when we take y=mx (general equation of all the lines passing through (0,0)) and substitute it in the equation we get x^2/(m^2 + x^2), this tends to "0" irrespective of the value of "m" suggesting the limit is zero along all the lines passing through (0,0), but when we take y=x^2 and substitute it in the equation, we get the limit as 1/2 and since it is different in value obtained from considering a path along a line, we say the limit doesn't exist.
So, can anyone explain (if possible geometrically) why we cant just evaluate limits by calculating the limits of all the possible lines passing through the point at which the limit is to be evaluated?
 A: Here's the picture my second year calculus teacher showed me that made me realize why you can't just check all the lines. Imagine a weirdly shaped cliff in the middle of the ocean:

The blue is the ocean, which is at sea level, and the green is the very long, very thin (infinitely thin, to be precise) cliff or island or whatever. It is at a height of, say, $100$ meters, and is in the shape of a parabola. The red point is right where the cliff ends and is the point we are interested in considering the limit of your altitude as you approach that point. We can clearly see from the diagram that if we approach the point using any straight line, we will think that the limit is sea level, or zero meters. But, if you walk along the top of the cliff, in that parabolic path, all the way right up to where the cliff ends, you won't agree with all the other observers - you will think that the limit is $100$ meters.
To make this more rigorous, consider the function
$$f(\boldsymbol x)=f(x_1,x_2)=\begin{cases}  1 & \text{if}~ x_1>0~\text{and}~x_2={x_1}^2  \\ 0 & \text{otherwise}\end{cases} $$
If I take any linear path, say
$$\boldsymbol x(t)=(t,a t)$$
Then
$$\lim_{t\to 0}f(\boldsymbol x(t)) = \lim_{t\to 0}f(t,at)$$
Since $x_2(t)=at\neq x_1(t)^2=t^2$ in general, we have $f(t,at)=0$ except at possibly finitely many points. So
$$\lim_{t\to 0}f(\boldsymbol x(t))=\lim_{t\to 0} 0=0.$$
However, if I take the parabolic path, i.e $\boldsymbol x(t)=(t,t^2)$ then
$$\lim_{t\to 0^+}f(\boldsymbol x(t))=\lim_{t\to 0^+}f(t,t^2)$$
Since $t>0$, we have $f(t,t^2)=1$ so this is
$$\lim_{t\to 0^+}f(\boldsymbol x(t))=\lim_{t\to 0^+}1=1$$
The limits aren't equal, thus the global limit
$$\lim_{\boldsymbol x\to (0,0)}f(\boldsymbol x)$$
Does not exist.
