Finding a function of two variables , satisfying $$\lim_{\left(x,y \right)\rightarrow \left({x}_{0},{y}_{0} \right)}f\left(x,y \right)=+\infty ;$$ and
for $\forall\delta>0, \exists y^{'},y^{''}\in (y_{0}-\delta,y_{0}+\delta)\setminus\lbrace y_{0}\rbrace,s.t. \lim_{x\rightarrow {x}_{0}}f\left(x,{y}^{'} \right)=+\infty,\lim_{x\rightarrow {x}_{0}}f\left(x,{y}^{''} \right)\in\mathbb{R}.$
I find a two variables function: $\lim_{\left(x,y \right)\rightarrow \left(0,0 \right)}\frac{1+x^{2}+y^{2}}{x^{2}}=+\infty,$ but $\forall\delta>0,$ all $y^{'} \in (-\delta,\delta)\setminus\lbrace 0 \rbrace, \lim_{x\rightarrow 0}f\left(x,{y}^{'} \right)=+\infty.$
Any of your help will be appreciated!
Consider $x_0,y_0=0$ and \begin{align*} f(x,y)=\begin{cases}\frac{1}{x^2+y^2}, & y>0 \\ \frac{1}{x^2}, & y\leq 0\neq x \\ \frac{1}{y^2}, & y<0=x \\ 0, & y=0=x\end{cases}. \end{align*} Then $f(x,y)\geq\min\{\frac{1}{x^2},\frac{1}{y^2}\}$ for each $(x,y)\neq(0,0)$ and hence $f(x,y)\to\infty$ as $(x,y)\to(0,0)$. Moreover, for $y<0\neq x$, $f(x,y)=1/x^2\to\infty$ as $x\to0$, and for $y>0$, $f(x,y)=\frac{1}{x^2+y^2}\to\frac{1}{y^2}\in\mathbb R$ as $x\to0$.