linguistic:translation
Multiple limits

Although less commonly used, there is another type of limit for a multivariable function, known as the multiple limit. For a two-variable function, this is the double limit. Let f: S × T → R {\displaystyle f:S\times T\to \mathbb {R} } be defined on S × T ⊆ R 2, {\displaystyle S\times T\subseteq \mathbb {R} ^{2},} we say the double limit of f as x approaches p and y approaches q is L, written

lim x → p y → q f (x, y) = L {\displaystyle \lim _{{x\to p} \atop {y\to q}}f(x,y)=L}

if the following condition holds:

(∀ ε > 0) (∃ δ > 0) (∀ x ∈ S) (∀ y ∈ T) ((0 < | x − p | < δ) ∧ (0 < | y − q | < δ) ⟹ | f (x, y) − L | < ε). {\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(\forall y\in T)\,((0<|x-p|<\delta)\land (0<|y-q|<\delta)\implies |f(x,y)-L|<\varepsilon).}

For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit. Consider the example f (x, y) = { 1 for x y ≠ 0 0 for x y = 0 {\displaystyle f(x,y)={\begin{cases}1\quad {\text{for}}\quad xy\neq 0\\0\quad {\text{for}}\quad xy=0\end{cases}}} where lim x → 0 y → 0 f (x, y) = 1 {\displaystyle \lim _{{x\to 0} \atop {y\to 0}}f(x,y)=1} but lim (x, y) → (0, 0) f (x, y) {\displaystyle \lim _{(x,y)\to (0,0)}f(x,y)} does not exist.

If the domain of f is restricted to (S ∖ { p }) × (T ∖ { q }), {\displaystyle (S\setminus \{p\})\times (T\setminus \{q\}),} then the two definitions of limits coincide.