Which of the following is the correct form of the ϵ\epsilonϵ-δ\deltaδ definition for limx→af(x)=L\lim_{x \to a} f(x) = Llimx→af(x)=L?
For all ϵ>0\epsilon > 0ϵ>0, there exists δ>0\delta > 0δ>0 such that 0<∣x−a∣<δ ⟹ ∣f(x)−L∣<ϵ0 < |x-a| < \delta \implies |f(x)-L| < \epsilon0<∣x−a∣<δ⟹∣f(x)−L∣<ϵ
There exists ϵ>0\epsilon > 0ϵ>0, for all δ>0\delta > 0δ>0, 0<∣x−a∣<δ ⟹ ∣f(x)−L∣<ϵ0 < |x-a| < \delta \implies |f(x)-L| < \epsilon0<∣x−a∣<δ⟹∣f(x)−L∣<ϵ
For all δ>0\delta > 0δ>0, there exists ϵ>0\epsilon > 0ϵ>0 such that ∣f(x)−L∣<ϵ ⟹ ∣x−a∣<δ|f(x)-L| < \epsilon \implies |x-a| < \delta∣f(x)−L∣<ϵ⟹∣x−a∣<δ
For all ϵ>0\epsilon > 0ϵ>0, there exists δ>0\delta > 0δ>0 such that ∣f(x)−L∣<ϵ ⟹ 0<∣x−a∣<δ|f(x)-L| < \epsilon \implies 0 < |x-a| < \delta∣f(x)−L∣<ϵ⟹0<∣x−a∣<δ