Find the limit of $$\exp\left(\frac{|x-2y|}{(x-2y)^2}\right)$$ when $(x,y) \to (2y,y)$.
I have considered two cases: $(x-2y)<0 $ and $(x-2y)>0$. But in first case the limit turns out to be $0$ and in the second case limit is undefined. I am not sure if my solution is correct or not.
It is :
$$\lim_{(x,y) \to (2y,y)} \exp\left({\frac{|x-2y|}{(x-2y)^2}}\right) = \lim_{(x,y) \to (2y,y)} \exp\left({\frac{|x-2y|}{|x-2y|^2}}\right)$$ $$=$$ $$\lim_{(x,y) \to (2y,y)} \exp\left({\frac{1}{|x-2y|}}\right) \equiv \lim_{z \to 0} \exp\left(\frac{1}{|z|} \right) = \infty$$
We have that by $t=x-2y \to 0$ we reduce to the simpler
$$\large e^{\frac{|x-2y|}{(x-2y)^2}}=e^{{|t|}/{t^2}}=e^{{1}/{|t|}}\to e^\infty=\infty$$
