How do you prove that a limit does not exist?

**gello88** · November 4th 2009, 05:32 PM

Hi i need help on how to prove a limit does not exist using the epsilon-delta definition of the limits.

Its like lim x--> 5 1/x-5 does not exist

how do i go about proving that this limit does not exist? and what do you think is a good approach to proving limits using the definition?

Thank you,
Angelo

**redsoxfan325** · November 4th 2009, 06:40 PM

Originally Posted by gello88

Hi i need help on how to prove a limit does not exist using the epsilon-delta definition of the limits.

Its like $\lim_{x\to5}\frac{1}{x-5}$ does not exist

how do i go about proving that this limit does not exist? and what do you think is a good approach to proving limits using the definition?

Thank you,
Angelo

This question was asked and answered earlier here.

**Bruno J.** · November 4th 2009, 06:42 PM

Yeah, seriously.

**tonio** · November 4th 2009, 06:50 PM

Originally Posted by gello88

Hi i need help on how to prove a limit does not exist using the epsilon-delta definition of the limits.

Its like lim x--> 5 1/x-5 does not exist

how do i go about proving that this limit does not exist? and what do you think is a good approach to proving limits using the definition?

Thank you,
Angelo

I'm not sure but I think you meant $\lim_{x\to 5}\frac{1}{x-5}$ ...so suppose the limit exists and equals L, then:

$\forall \epsilon >0\,\,\exists N_\epsilon \in \mathbb{N}\,\,s.t.\,\,\forall\,\,n>N_\epsilon\,,\, \,\mid \frac{1}{x-5}-L\mid <\epsilon \Longleftrightarrow L-\epsilon <\frac{1}{x-5}< L+\epsilon$ (*)

On the other hand, the expression $\frac{1}{x-5}$ can be made as big (as negative big) as wanted by choosing x close enough to 5 from the right (from the left): $\frac{1}{x-5}>K\longrightarrow x-5<\frac{1}{K}$ , and this contradicts (*) above since there $\frac{1}{x-5}$ is bounded...

- If $L=\infty\,(-\infty)$ the proof is easier, by making x approach 5 from the left (from the right).

- Tonio

**gello88** · November 4th 2009, 06:58 PM

ouu thats right, sorry about that.

But the answer there was handled using one sided limits which I know is a very good way of proving it and it works. But my question is how do i go about proving this limit does not exist using the:

epsilon - delta definition?

∀ε > 0, ∃δ > 0 such that 0<|x-c|<δ ---> |f(x) - L| < ε

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