information:extraction, structural:summarization
Ordinary limits

By noting that | x − p | represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function f: S × T → R {\displaystyle f:S\times T\to \mathbb {R} } defined on S × T ⊆ R 2, {\displaystyle S\times T\subseteq \mathbb {R} ^{2},} we defined the limit as follows: the limit of f as (x, y) approaches (p, q) is L, written

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

if the following condition holds:

or formally: (∀ ε > 0) (∃ δ > 0) (∀ x ∈ S) (∀ y ∈ T) (0 < (x − p) 2 + (y − q) 2 < δ ⟹ | f (x, y) − L | < ε)). {\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(\forall y\in T)\,(0<{\sqrt {(x-p)^{2}+(y-q)^{2}}}<\delta \implies |f(x,y)-L|<\varepsilon)).}

Here (x − p) 2 + (y − q) 2 {\textstyle {\sqrt {(x-p)^{2}+(y-q)^{2}}}} is the Euclidean distance between (x, y) and (p, q). (This can in fact be replaced by any norm | | (x, y) − (p, q) | |, and be extended to any number of variables.)

For example, we may say lim (x, y) → (0, 0) x 4 x 2 + y 2 = 0 {\displaystyle \lim _{(x,y)\to (0,0)}{\frac {x^{4}}{x^{2}+y^{2}}}=0} because for every ε > 0, we can take δ = ε {\textstyle \delta ={\sqrt {\varepsilon }}} such that for all real x ≠ 0 and real y ≠ 0, if 0 < (x − 0) 2 + (y − 0) 2 < δ, {\textstyle 0<{\sqrt {(x-0)^{2}+(y-0)^{2}}}<\delta,} then | f (x, y) − 0 | < ε.

Similar to the case in single variable, the value of f at (p, q) does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q). In the above example, the function f (x, y) = x 4 x 2 + y 2 {\displaystyle f(x,y)={\frac {x^{4}}{x^{2}+y^{2}}}} satisfies this condition. This can be seen by considering the polar coordinates (x, y) = (r cos θ, r sin θ) → (0, 0), {\displaystyle (x,y)=(r\cos \theta,r\sin \theta)\to (0,0),} which gives lim r → 0 f (r cos θ, r sin θ) = lim r → 0 r 4 cos 4 θ r 2 = lim r → 0 r 2 cos 4 θ. {\displaystyle \lim _{r\to 0}f(r\cos \theta,r\sin \theta)=\lim _{r\to 0}{\frac {r^{4}\cos ^{4}\theta }{r^{2}}}=\lim _{r\to 0}r^{2}\cos ^{4}\theta.} Here θ = θ (r) is a function of r which controls the shape of the path along which f is approaching (p, q). Since cos θ is bounded between, by the sandwich theorem, this limit tends to 0.

In contrast, the function f (x, y) = x y x 2 + y 2 {\displaystyle f(x,y)={\frac {xy}{x^{2}+y^{2}}}} does not have a limit at (0, 0). Taking the path (x, y) = (t, 0) → (0, 0), we obtain lim t → 0 f (t, 0) = lim t → 0 0 t 2 = 0, {\displaystyle \lim _{t\to 0}f(t,0)=\lim _{t\to 0}{\frac {0}{t^{2}}}=0,} while taking the path (x, y) = (t, t) → (0, 0), we obtain lim t → 0 f (t, t) = lim t → 0 t 2 t 2 + t 2 = 1 2. {\displaystyle \lim _{t\to 0}f(t,t)=\lim _{t\to 0}{\frac {t^{2}}{t^{2}+t^{2}}}={\frac {1}{2}}.}

Since the two values do not agree, f does not tend to a single value as (x, y) approaches (0, 0).