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If $file$ is an actual-valued function outlined with a established $Esubset mathbb R$ (or $subset mathbb R^k$), the upper and lower limits of $f$ at $x_0$ are denoted bythen the inequalities previously mentioned become equalities (with lim sup n → ∞ a n displaystyle limsup _ nto infty a_ n adjustments signal, that means which the signal on th