Editing Limit of a function (section)

===Ordinary limits===
By noting that {{math|{{abs|''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 <math>f : S \times T \to \R</math> defined on <math>S \times T \subseteq \R^2,</math> we defined the limit as follows: '''the limit of {{mvar|f}} as {{math|(''x'', ''y'')}} approaches {{math|(''p'', ''q'')}} is {{mvar|L}}''', written

<math display=block> \lim_{(x,y) \to (p, q)} f(x, y) = L </math>

if the following condition holds:
:For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that for all {{mvar|x}} in {{mvar|S}} and {{mvar|y}} in {{mvar|T}}, whenever <math display=inline>0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta,</math> we have {{math|{{abs|''f''(''x'', ''y'') − ''L''}} < ''ε''}},<ref>{{citation
 | last = Stewart | first = James |author-link = James Stewart (mathematician)
 | chapter = Chapter 14.2 Limits and Continuity
 | pages = 952
 | title = Multivariable Calculus
 | year = 2020
 | publisher = Cengage Learning | edition = 9th
 | isbn = 9780357042922}}</ref>

or formally:
<math display=block>(\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)).</math>

Here <math display=inline>\sqrt{(x-p)^2 + (y-q)^2}</math> is the [[Euclidean distance]] between {{math|(''x'', ''y'')}} and {{math|(''p'', ''q'')}}. (This can in fact be replaced by any [[Norm (mathematics)|norm]] {{math|{{abs|{{abs|(''x'', ''y'') − (''p'', ''q'')}}}}}}, and be extended to any number of variables.)

For example, we may say
<math display=block> \lim_{(x,y) \to (0, 0)} \frac{x^4}{x^2+y^2} = 0 </math>
because for every {{math|''ε'' > 0}}, we can take <math display=inline>\delta = \sqrt \varepsilon</math> such that for all real {{math|''x'' ≠ 0}} and real {{math|''y'' ≠ 0}}, if <math display=inline>0 < \sqrt{(x-0)^2 + (y-0)^2} < \delta,</math> then {{math|{{abs|''f''(''x'', ''y'') − 0}} < ''ε''}}.

Similar to the case in single variable, the value of {{mvar|f}} at {{math|(''p'', ''q'')}} does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of {{mvar|f}} approaches {{mvar|L}} along every possible path approaching {{math|(''p'', ''q'')}}.{{sfnp|Stewart|2020|p=953}} In the above example, the function
<math display=block>f(x, y) = \frac{x^4}{x^2+y^2}</math>
satisfies this condition. This can be seen by considering the [[polar coordinates]] 
<math display=block>(x,y) = (r\cos\theta, r\sin\theta) \to (0, 0),</math>
which gives
<math display=block>\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.</math>
Here {{math|1=''θ'' = ''θ''(''r'')}} is a function of ''r'' which controls the shape of the path along which {{mvar|f}} is approaching {{math|(''p'', ''q'')}}. Since {{math|cos ''θ''}} is bounded between [−1, 1], by the [[sandwich theorem]], this limit tends to 0.

In contrast, the function
<math display=block>f(x, y) = \frac{xy}{x^2 + y^2}</math>
does not have a limit at {{math|(0, 0)}}. Taking the path {{math|1=(''x'', ''y'') = (''t'', 0) → (0, 0)}}, we obtain
<math display=block>\lim_{t \to 0} f(t, 0) = \lim_{t \to 0} \frac{0}{t^2} = 0,</math>
while taking the path {{math|1=(''x'', ''y'') = (''t'', ''t'') → (0, 0)}}, we obtain
<math display=block>\lim_{t \to 0} f(t, t) = \lim_{t \to 0} \frac{t^2}{t^2 + t^2} = \frac{1}{2}.</math>

Since the two values do not agree, {{mvar|f}} does not tend to a single value as {{math|(''x'', ''y'')}} approaches {{math|(0, 0)}}.