Remember the Limits Formalized Definition If we have a limit like: Then:
Rough Work
We would then find a value such that , Rearrange the epsilon statement 2|x-3|<\epsilon\ Then, we can say that
Proof
Let be arbitrary Choose Assume Assume Now, Thus,
Remember the Limits Formalized Definition ∀ϵ>0,∃δ>0 such that 0<∣x−c∣<δ⟹∣f(x)−L∣<ϵ If we have a limit like: limx→3(−2x+1)=−5 Then:
We would then find a value δ such that ∣x−3∣<δ, ∣−2x+1−(−5)∣<ϵ Rearrange the epsilon statement ∣−2x+6∣<ϵ 2|x-3|<\epsilon\ ∣x−3∣<2ϵ Then, we can say that δ=2ϵ
Let ϵ>0 be arbitrary Choose δ=2ϵ Assume ∣x−3∣<δ Assume ∣−2x+1−(−5)∣<ϵ Now, ∣−2x+1−(−5)∣=∣−2x+6∣=2∣x−3∣<22ϵ 2∣x−3∣<ϵ Thus, ∣(−2x+1)−(−5)∣<ϵ □