Is an expression such as the following mathematically admissible?
$$\lim_{h(x) \to h_1} f(x)$$
where $f, h: X \subset \mathbb{R}^n \to \mathbb{R}$ are some continuous functions and $h_1 \in \mathbb{R}$.
Motivating examples are
$$\lim_{x + y \to 0} (x+y) = 0$$
and
$$\lim_{x^2 + y^2 \searrow 0} (x^2 + 2 y^2 + y ) = 0$$
Thank you.
I learned topology and logic recently. I tried to apply my new knowledge to this question.
Let $f: X \rightarrow \mathbb{R}$ and $h: X \rightarrow \mathbb{R}^n$. For some $\alpha \in \mathbb{R}^n$ and $l \in \mathbb{R}$, $$ \lim_{h(x) \rightarrow \alpha} f(x) = l $$ is equivalent to $$ \begin{align} &\forall \epsilon > 0 :\exists\delta > 0 : \forall x \in X:\Vert h(x) -\alpha\Vert < \delta \Rightarrow |f(x)-l| < \epsilon\\ &\forall \epsilon > 0 :\exists\delta > 0 : \forall x \in X: h(x) \in B_\delta(\alpha) \Rightarrow f(x) \in B_\epsilon(l)\\ &\forall \epsilon > 0 :\exists\delta > 0 : \forall x \in X: x \in h^{-1}[B_\delta(\alpha)] \Rightarrow x \in f^{-1}[B_\epsilon(l)]\\ &\forall \epsilon > 0 :\exists\delta > 0 : h^{-1}[B_\delta(\alpha)] \subseteq f^{-1}[B_\epsilon(l)] \end{align} $$ If we can find such $\delta$ for each $\epsilon$, whatever $f$ and $h$ are, we can conclude that $f(x)$ converges to $l$.
This question was originally posted after seeing the notation in Wikipedia