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Definition: “The Limit of a Function”

Let f be a function defined at least on some set of the form (c-p, c) U (c, c+p). $\displaystyle\underset{{x\to c}}{\mathop{{\lim }}}\, f(x)=l$ if for each $\displaystyle \varepsilon>0$ there exists $\displaystyle \delta >0$ such that if $\displaystyle 0<\left| {x-c} \right|<\delta $ for $\displaystyle\delta $ , then $\displaystyle \left| {f(x)-l} \right|<\varepsilon$