linguistic:paraphrasing, information:filtering, quality:correction, linguistic:voice_change
Euclidean metric

The limit in Euclidean space is a direct generalization of limits to vector-valued functions. For example, we may consider a function f: S × T → R 3 {\displaystyle f:S\times T\to \mathbb {R} ^{3}} such that f (x, y) = (f 1 (x, y), f 2 (x, y), f 3 (x, y)). {\displaystyle f(x,y)=(f_{1}(x,y),f_{2}(x,y),f_{3}(x,y)).} Then, under the usual Euclidean metric, lim (x, y) → (p, q) f (x, y) = (L 1, L 2, L 3) {\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=(L_{1},L_{2},L_{3})} if the following holds:

(∀ ε > 0) (∃ δ > 0) (∀ x ∈ S) (∀ y ∈ T) (0 < (x − p) 2 + (y − q) 2 < δ ⟹ (f 1 − L 1) 2 + (f 2 − L 2) 2 + (f 3 − L 3) 2 < ε). {\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(\forall y\in T)\,\left(0<{\sqrt {(x-p)^{2}+(y-q)^{2}}}<\delta \implies {\sqrt {(f_{1}-L_{1})^{2}+(f_{2}-L_{2})^{2}+(f_{3}-L_{3})^{2}}}<\varepsilon \right).}

In this example, the function concerned are finite- dimension vector-valued function. In this case, the limit theorem for vector-valued function states that if the limit of each component exists, then the limit of a vector-valued function equals the vector with each component taken the limit: lim (x, y) → (p, q) (f 1 (x, y), f 2 (x, y), f 3 (x, y)) = (lim (x, y) → (p, q) f 1 (x, y), lim (x, y) → (p, q) f 2 (x, y), lim (x, y) → (p, q) f 3 (x, y)). {\displaystyle \lim _{(x,y)\to (p,q)}{\Bigl (}f_{1}(x,y),f_{2}(x,y),f_{3}(x,y){\Bigr)}=\left(\lim _{(x,y)\to (p,q)}f_{1}(x,y),\lim _{(x,y)\to (p,q)}f_{2}(x,y),\lim _{(x,y)\to (p,q)}f_{3}(x,y)\right).}