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the limit of $ f(x) $ as $x$ approaches $ x_0 $ is \textbf{the number} $ L $

if the folowwing criterion holds:

giver any radius $ \epsilon > 0 $ about $ L $, there exist radius 

$ \delta $ about $ x_0 $ such that for all $x$,

\[0<|x-x_0|<\delta \mbox { implies } |f(x)-L|<\epsilon\]

if a function $ f(x) $ has a limit $ L $ as $x$ approaches $x_0$, its denoted by

\[ \lim\limits_{x \to x_0} f(x)=L\]