Limit of $f$ at $(0,0)$ vs limits of $f(x,kx)$ when $x\to0$ Why if for all $k\in\mathbb R$ we have $f(x,kx)\to\ell$ when $x\to 0$ then this is not enough to conclude that $f(x,y)\to\ell$ if $(x,y)\to (0,0)$? 

Indeed, if the limit is the same when we can consider all straight lines that pass through $(0,0)$, logically it should be enough to conclude on the limit, but I found a counter example. So why doesn't it work ? What is the mathematical reason behind this phenomenon?
For a counterexample, let $f(x,y)=\frac{x^2y}{x^4+y^2}$. Then $f(x,kx)=\frac{kx^3}{x^4+k^2x^2}=\frac{kx}{x^2+k^2}\to 0$ as $x\to 0$. But $f(x,x^2)=\frac{x^4}{2x^4}=\frac{1}{2}\not\to 0$

The idea is that $y$ can approach $0$ faster than $x$ can by a LOT, giving issues.
For a function in more than one  variable;

For limit of a  function $f$   to exist at a particular point we have to consider all possible paths for reaching that point and see whether the limit exits or not. What you are considering are only  the straight line paths.

There can be many ways to reach that point