Question: > 
> Find limit of $ \underset{\{x,y\}\to \{0,0\}}{\text{lim}}\frac{-\frac{x y}{2}+\sqrt{x y+1}-1}{y \sqrt{x^2+y^2}}$
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> 
> 

How can I do that? It is interesting due to mathematica says that 
$$\underset{\{x,y\}\to \{0,0\}}{\text{lim}}\frac{-\frac{x y}{2}+\sqrt{x y+1}-1}{y \sqrt{x^2+y^2}} = 0 $$
but wolfram that limit doesn't exists. What is more I am not sure too about existance of limit due to
$$\underset{\{x,y\}\to \{0,0\}}{\text{lim}}\frac{\sqrt{x y+1}-1}{y \sqrt{x^2+y^2}} $$ doesn't exists too...

Answer: 
You are hinted by the fact that if the pieces have equal area, they must have sides $\dfrac{\sqrt5}2$, which evokes the diagonal of $2\times1$ rectangles.

[![enter image description here](https://i.stack.imgur.com/p64vV.png)](https://i.stack.imgur.com/p64vV.png)
This image illustrates how the requisite dissection arises naturally:

[![enter image description here](https://i.stack.imgur.com/wFxmL.gif)](https://i.stack.imgur.com/wFxmL.gif)