The important thing is that a limit cannot depend on the direction we take to reach a point. If it does, the function is not continuous.
Let’s say that we have a function $f(x,y)$ and we want to find the limit as $(x,y) \to (0,0)$. One way we can do this is to follow a line and see what limit we reach when we follow it, for example the line $y=0$.
Let’s say $f(x,y) = 4x^2y + 2x - 3$. We can follow the line $y=0$, in which case:
$$ f(x) = 2x - 3 $$
This becomes single-variable limit and it’s clearly $-3$ when $x=0$. Now let’s pick another line, $y = x$:
$$ f(x) = 4x^4 + 2x - 3 $$
It’s the same case: $-3$. Let’s try one more weird one, for fun - $y=e^x-1$:
$$ f(x) = 4x^2(e^x-1) + 2x - 3 $$
Again, the value at $x=0$ is just $-3$.
Let’s take an example of a discontinuous function:
$$ f(x,y) = \frac{4\sin (xy)}{x^2+y^2} - 5 $$
Let’s follow along $x=0$:
$$ f(y) = \frac{0}{y^2} - 5 $$
with the exception of $y^2=0$, if we follow $x=0$, the value is $-5$ so the limit is also $-5$.
Now let’s follow $y=x$: