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The limit of a function ff at x0x_0 is LL iff limโกxโ†’x0f(x)=L{\lim\limits_{x \to x_0}} f (x) = L iff

โˆ€ฯต>0,โˆƒฮด>0:โˆ€x:0<โˆฃxโˆ’x0โˆฃ<ฮดโ†’โˆฃf(x)โˆ’Lโˆฃ<ฯต\forall \epsilon > 0, \exists \delta >0 : \forall x : 0 < |x - x_0| < \delta \to |f(x) - L| < \epsilon

ff does not need to exist at x0x_0 to define the limit. However, for the function to be continuous, the limโกxโ†’x0f(x)=f(x0)\lim\limits_{x \to x_0} f(x) = f(x_0).

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