Definition
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[20D]Let \(I⊂ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), \(f:I→ ℝ\) function, \(l∈\overline{ℝ}\).
The idea of limit (right or left or bilateral) is thus expressed.
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\(\lim _{x→ x_ 0} f(x) = l\) |
for every ”full” neighbourhood \(V\) of \(l\), there exists a “deleted” neighbourhood \(U\) of \(x_ 0\) such that for every \(x∈ U∩ I\), you have \(f(x)∈ V\) |
where the neighborhood \(U\) will be “right” or “left’ if the limit is “right” or “left”; it can also be said that
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\(\lim _{x→ x_ 0} f(x) = l\) |
for every ”full” neighbourhood \(V\) of \(l\), you have \(f(x)∈ V\) eventually for \(x\) tending to \(x_ 0\) |
adding that \(x{\gt}x_ 0\) if the limit is “right”, or \(x{\lt}x_ 0\) if the limit is “left”.