Given a function $f$ defined on the set $\Omega=\{(x,y)\in \mathbb R^2 \mid x\neq 1\}$. Does this limit exist?
$$ \lim_{(x,y)\to(1,0)}f(x,y)=\lim_{(x,y)\to(1,0)}\frac{y}{x-1} $$
I procede in this way taking these 2 lines $$ \{ (x,y)\mid y=0, x\neq1 \} $$ the limit here should be $0$. While here $$y=x-1$$ the limit is $1$. This line goes through the point $(1,0)$. Therefore the limit doesn't exist. But why?
Should every line/function cross the point inserted in the $f$ function to have the same limit?
What is the visual explanation of what is happening?
As you noticed if we consider the following paths
$x\neq 1$, $y=0$
$x=1+t$, $y=t$ with $t\to 0$
we obtain to different values for the limit.
Now it can be shown that when the limit exist it is unique therefore whenever we can find different paths with different limit we can conclude that the limit doesn't exist.