Mathematics

Mathematics

Problem

Which of the following statements correctly defines lim⁡x→cf(x)=L\displaystyle \lim_{x\to c}f(x)=L? a) There exists δ>0\delta>0 such that for every ε>0\varepsilon>0, if 0<∣x−c∣<δ0<|x-c|<\delta then ∣f(x)−L∣<ε|f(x)-L|<\varepsilon. b) For every δ>0\delta>0 there exists ε>0\varepsilon>0 such that if 0<∣x−c∣<δ0<|x-c|<\delta then ∣f(x)−L∣<ε|f(x)-L|<\varepsilon. c) For every ε>0\varepsilon>0 there exists δ>0\delta>0 such that if 0<∣x−c∣<δ0<|x-c|<\delta then ∣f(x)−L∣<ε|f(x)-L|<\varepsilon. d) For every ε>0\varepsilon>0 there exists δ>0\delta>0 such that if ∣f(x)−L∣<ε|f(x)-L|<\varepsilon then 0<∣x−c∣<δ0<|x-c|<\delta.

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