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Proof that Lim doesn't exist

  1. Oct 20, 2016 #1
    1. The problem statement, all variables and given/known data
    Proof that the limit of the function below doesn't exists.
    limx-->1 1/(x-1)


    2. Relevant equations


    3. The attempt at a solution
    Lets assume that limit L exists.
    So if (1) 0< |x-1| < δ then (2) |1/(x-1) - L| < ε

    at the book they gave an example by giving a value to ε.
    put ε = 1. then showing a contradiction by giving two δ values to x.

    but now im thinking about what values can i put that satisfy (1) that for them |1/(x-1) - L| < 1 doesn't hold.
     
    Last edited by a moderator: Oct 20, 2016
  2. jcsd
  3. Oct 20, 2016 #2
    Maybe if ε = L/2

    then if i put x1= δ/2+1, x2 = -δ/2 +1 both of them satisfy (1)
    then i get for (2)
    |δ/2 - L| < L/2, and |-δ/2 - L| < L/2 δ>0
    and we can see that |-δ/2 - L| < L/2 doesn't hold. and that is contradiction, and therefore the limit doesn't exists.
     
  4. Oct 20, 2016 #3
    now i see i haven't shown that L ≠ 0, because ε > 0.


    ε = δ/2, looks like it would work now. since δ >0.

    |δ/2 - L| < δ/2, and |-δ/2 - L| < δ/2

    and in all cases of L, L>0, L<0. L=0 there are contradictions.
    and therefore there is no limit.
     
    Last edited: Oct 20, 2016
  5. Oct 20, 2016 #4

    Mark44

    Staff: Mentor

    Instead of a proof by contradiction, why don't you try proving this directly? Looking at the graph of f(x) = 1/(x - 1), it's clear that the limit doesn't exist (in any sense), because ##\lim_{x \to 1^-}\frac 1 {x - 1} = -\infty## while ##\lim_{x \to 1^+}\frac 1 {x - 1} = \infty##

    If you can use the definition of the limit to prove each of these one-sided limits, that should do the job.
     
  6. Oct 20, 2016 #5
    Sorry, yes you are right, i can do that too.
     
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