For the problem :
A part of the solution was given as :
We now claim that $\lim\limits_{(x, y) \to(0,0)} f(x, y)$ does not exist. Assume to the contrary that $\lim\limits_{(x, y)\to(0,0)} f(x, y)$ exists and is equal to a real number, say $L .$ In that case, for $\varepsilon=1$ there exists $\delta>0$ such that $\left|\frac{4 x(4 x+y)}{\sqrt{|4 x|}-\sqrt{|y|}}-L\right|<1 \quad$ whenever $(x, y) \in D$ is such that $0<\|(x, y)\|<\delta$ Now, choose and fix $x=\delta / 8$ and a positive integer $N$ such that $N \geq \sqrt{2 / \delta}$ For any $n \geq N,$ if we consider $y_{n}$ satisfying $\sqrt{y_n}=\sqrt{4 x}-\frac1n$ then it is straightforward to verify that $\left(x, y_n\right) \in D, 0<\left\|\left(x, y_n\right)\right\|<\delta$ and $$ f\left(x, y_n\right)=\frac{4 x\left(4 x+y_n\right)}{\sqrt{|4 x|}-\sqrt{|y|}}=4 n x\left(4 x+y_n\right) \geq 16 n x^2=n \delta^2/ 4 $$ But then $f\left(x, y_n\right) \to \infty$ as $n \to \infty$ and hence it is not possible to have that $\left|f\left(x, y_n\right)-L\right|<$ 1 for all $n \geq N$ for any real number $L$ Thus, we conclude that $\lim\limits_{(x, y)\to(0,0)} f(x, y)$ does not exist.
Are the choices of ε, δ, N and y(n) arbitrary? Are all these choices based on logic or is it just "intuition"? Is it possible to prove the same without using so many assumptions?