Editing Limit of a function (section)

===Limits at infinity===
{{^|[[Limit at infinity]] redirects here}}
[[File:Limit Infinity SVG.svg|thumb|300px|The limit of this function at infinity exists]]

Let <math>f:S \to \R</math> be a function defined on <math>S \subseteq \R.</math> '''The limit of {{mvar|f}} as {{mvar|x}} approaches infinity is {{mvar|L}}''', denoted

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

means that:
{{block indent| For every {{math|''ε'' > 0}}, there exists a {{math|''c'' > 0}} such that whenever {{math|+''x'' > ''c''}}, we have {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0 )\, (\exists c > 0) \,(\forall x \in S) \,(x > c \implies |f(x) - L| < \varepsilon).</math>

Similarly, '''the limit of {{mvar|f}} as {{mvar|x}} approaches minus infinity is {{mvar|L}}''', denoted

<math display=block> \lim_{x \to -\infty}f(x) = L,</math>

means that:
{{block indent|For every {{math|''ε'' > 0}}, there exists a {{math|''c'' > 0}} such that whenever {{math|''x'' < −''c''}}, we have {{math|{{abs|''f''(''x'') − ''L''}} < ''ε''}}.}}
<math display=block>(\forall \varepsilon > 0)\, (\exists c > 0) \,(\forall x \in S)\, (x < -c \implies |f(x) - L| < \varepsilon).</math>

For example, 
<math display=block> \lim_{x \to \infty} \left(-\frac{3\sin x}{x} + 4\right) = 4</math>
because for every {{math|''ε'' > 0}}, we can take {{math|1=''c'' = 3/''ε''}} such that for all real {{mvar|x}}, if {{math|''x'' > ''c''}}, then {{math|{{abs|''f''(''x'') − 4}} < ''ε''}}.

Another example is that
<math display=block> \lim_{x \to -\infty}e^{x} = 0</math>
because for every {{math|''ε'' > 0}}, we can take {{math|1=''c'' = max{1, −ln(''ε'')} }} such that for all real {{mvar|x}}, if {{math|''x'' < −''c''}}, then {{math|{{abs|''f''(''x'') − 0}} < ''ε''}}.