prof-if-the-limitf-x-L-and-limit-f-x-M-then-L-M- Tinku Tara June 4, 2023 None 0 Comments FacebookTweetPin Question Number 116939 by joki last updated on 08/Oct/20 profifthelimitf(x)=Landlimitf(x)=M,thenL=M Answered by MAB last updated on 08/Oct/20 limx→x0f(x)=L=M⇒[(∀ε>0)(∃α>0)(∀x∈Df):∣x−x0∣<α⇒∣f(x)−L∣<ε]and[(∀ε′>0)(∃α′>0)(∀x∈Df):∣x−x0∣<α′⇒∣f(x)−M∣<ε′]wehave∣M−L∣⩽∣f(x)−L∣+f(x)−M∣triangleinequalityhence(∀ϵ>0)(∃δ=min(α,α′)>0):∣x−x0∣<δ⇒∣M−L∣<∍finallyM=L Answered by MAB last updated on 08/Oct/20 limx→x0f(x)=L=M⇒[(∀ε>0)(∃α>0)(∀x∈Df):∣x−x0∣<α⇒∣f(x)−L∣<ε]and[(∀ε′>0)(∃α′>0)(∀x∈Df):∣x−x0∣<α′⇒∣f(x)−M∣<ε′]wehave∣M−L∣⩽∣f(x)−L∣+f(x)−M∣triangleinequalityhence(∀ϵ>0)(∃δ=min(α,α′)>0):∣x−x0∣<δ⇒∣M−L∣<∍finallyM=L Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Given-f-x-5-x-find-f-x-by-using-limit-first-principal-derivative-Next Next post: If-you-walk-around-a-triangle-4m-each-side-such-a-way-that-you-keep-a-distance-of-2m-from-it-then-how-much-distance-will-you-travel- Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.
![lim_(x→x_0 ) f(x)=L=M ⇒[(∀ε>0)(∃α>0)(∀x∈D_f ):∣x−x_0 ∣<α⇒∣f(x)−L ∣<ε] and [(∀ε′>0)(∃α′>0)(∀x∈D_f ):∣x−x_0 ∣<α′⇒∣f(x)−M ∣<ε′] we have ∣M−L∣≤∣f(x)−L∣+f(x)−M∣ triangle inequality hence (∀ε>0)(∃δ=min(α,α′)>0): ∣x−x_0 ∣<δ⇒∣M−L∣<∍ finally M=L](https://www.tinkutara.com/question/Q116944.png)
![lim_(x→x_0 ) f(x)=L=M ⇒[(∀ε>0)(∃α>0)(∀x∈D_f ):∣x−x_0 ∣<α⇒∣f(x)−L ∣<ε] and [(∀ε′>0)(∃α′>0)(∀x∈D_f ):∣x−x_0 ∣<α′⇒∣f(x)−M ∣<ε′] we have ∣M−L∣≤∣f(x)−L∣+f(x)−M∣ triangle inequality hence (∀ε>0)(∃δ=min(α,α′)>0): ∣x−x_0 ∣<δ⇒∣M−L∣<∍ finally M=L](https://www.tinkutara.com/question/Q116945.png)