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Simple proof question (limits)

  1. Jul 23, 2014 #1
    I was reviewing some of the material in my calculus textbook when I realized that I really struggle with writing proofs. Therefore, I decided to attempt one of the easier proof related exercises found in the book. I would ask my teacher for help, but I am off for summer break.
    1. The problem statement, all variables and given/known data
    Prove that if the limit x->c of f(x) exists and limit x->c of [f(x)+g(x)] does not exist, then limit x->c of g(x) does not exist


    2. Relevant equations


    3. The attempt at a solution
    Based on the properties of limits, if limit x→c of f(x)=L and limit x→c of g(x)=K, then the limit x→c of [f(x)+g(x)]=L+K. So let us assume that the limit x→c of g(x) does exist and is equal to K, and the limit x→c f(x) is equal to L. If that is so, then limit x→c of [f(x)+g(x)] must exist and is equal to L+K. However this creates a contradiction, as it was stated that limit x→c of [f(x)+g(x)] does not exist and therefore ≠ L+K. Therefore lim x→c of g(x) cant exist, else a contradiction is created.

    So I tried to do a "proof by contradiction"... was I successful? If not, where exactly did I go wrong? Thanks
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 23, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, that is exactly right.
     
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