Editing Limit of a function (section)

===Existence and one-sided limits===
{{Main|One-sided limit}}
[[Image:Upper semi.svg|thumb|The limit as <math>x \to x_0^+</math> differs from that as <math>x \to x_0^-.</math> Therefore, the limit as {{math|''x'' → ''x''<sub>0</sub>}} does not exist.]]

Alternatively, {{mvar|x}} may approach {{mvar|p}} from above (right) or below (left), in which case the limits may be written as

<math display=block> \lim_{x \to p^+}f(x) = L </math>

or

<math display=block> \lim_{x \to p^-}f(x) = L </math>
[[File:Undefined limit examples.png|thumb|The first three functions have points for which the limit does not exist, while the function<math display="block"> f(x) = \frac{\sin(x)}{x} </math>is not defined at <math>x = 0</math>, but its limit does exist.]]
respectively. If these limits exist at p and are equal there, then this can be referred to as ''the'' limit of {{math|''f''(''x'')}} at {{mvar|p}}.{{sfnp|Swokowski|1979|p=72–73}} If the one-sided limits exist at {{mvar|p}}, but are unequal, then there is no limit at {{mvar|p}} (i.e., the limit at {{mvar|p}} does not exist). If either one-sided limit does not exist at {{mvar|p}}, then the limit at {{mvar|p}} also does not exist.

A formal definition is as follows. The '''limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} from above is {{mvar|L}}''' if:
:For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that whenever {{math|0 < ''x'' − ''p'' < ''δ''}}, we have {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.
<math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a,b))\, (0 < x - p < \delta \implies |f(x) - L| < \varepsilon).</math>

The '''limit of {{mvar|f}} as {{mvar|x}} approaches {{mvar|p}} from below is {{mvar|L}}''' if:
:For every {{math|''ε'' > 0}}, there exists a {{math|''δ'' > 0}} such that whenever {{math|0 < ''p'' − ''x'' < ''δ''}}, we have {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.
<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in (a,b)) \, (0 < p - x < \delta \implies |f(x) - L| < \varepsilon).</math>

If the limit does not exist, then the [[Oscillation of a function at a point|oscillation]] of {{mvar|f}} at {{mvar|p}} is non-zero.