# Homework Help: Proof that Lim doesn't exist

1. Oct 20, 2016

### Dank2

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. Oct 20, 2016

### Dank2

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.

3. Oct 20, 2016

### Dank2

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
4. Oct 20, 2016

### 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.

5. Oct 20, 2016

### Dank2

Sorry, yes you are right, i can do that too.