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Suppose that $g,h : \mathbb{R} \rightarrow \mathbb{R}^+$ and $f: \mathbb{R}^2 / \left\{(0,0)\right\} \rightarrow \mathbb{R}^+$ and that $g(x_1) \le f(x_1,x_2) \le h(x_1)$ where not both $x_1,x_2$ are zero and suppose that $\lim_{x_1 \rightarrow 0}g(x_1) = \lim_{x_1 \rightarrow 0} h(x_1)=L$. Can we then conclude that $\lim_{x_1 \rightarrow 0, x_2 \rightarrow 0} f(x_1,x_2) = L$?

In particular, does there exist a generalized Squeeze/Sandwich Theorem for functions of more than one variable?

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Yes, you can, because if $(x_1,x_2)\to0$, then $x_1\to0$.

That said, your inequality is a very strong condition, if it works for any $x_2$. It implies in particular that $f(0,x_2)=L$ for all $x_2$.

The Squeeze property has nothing to do with the number of variables. It just uses the fact that if $|f(x)|\leq |g(x)|$ and $g(x)\to0$, then $f(x)\to0$. Here $x$ can be a single variable or an $n$-tuple, it makes no difference.