# Uniqueness of the Limit.

• December 23rd 2005, 10:29 AM
ThePerfectHacker
Uniqueness of the Limit.
I was thinking perhaps the limit of a function at a constant or at infinity is not unique. Thus by delta-epsilon I mean there exists another number L' such that satisfies the condition of the limit. Thus, prove that if the limit exists it is unique. I was able to show that if L is one limit and L' is another limit then (L+L')/2 is also a limit. Thus, this shows that if a limit is not unique there exists infinitely many limits for that function. But this is not true I was trying to proof that because my books on advanced calculus did not consider this concept.
• December 23rd 2005, 12:06 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
I was thinking perhaps the limit of a function at a constant or at infinity is not unique. Thus by delta-epsilon I mean there exists another number L' such that satisfies the condition of the limit. Thus, prove that if the limit exists it is unique. I was able to show that if L is one limit and L' is another limit then (L+L')/2 is also a limit. Thus, this shows that if a limit is not unique there exists infinitely many limits for that function. But this is not true I was trying to proof that because my books on advanced calculus did not consider this concept.

The limit of a real function $f(x)$ as $x \rightarrow a$ if it exists is unique. Suppose otherwise, then there exist $L_1$ and $L_2$ such that given any $\varepsilon >0$ there exist $\delta_1>0$ and $\delta_2>0$ such that when:

$|x-a|< \delta_1\ \Rightarrow \ |f(x)-L_1|<\varepsilon$

and:

$|x-a|< \delta_2\ \Rightarrow \ |f(x)-L_2|<\varepsilon$.

So let $\delta=min(\delta_1,\delta_2)$, then we have:

$|x-a|< \delta\ \Rightarrow \ |f(x)-L_1|<\varepsilon$,

and

$|x-a|< \delta\ \Rightarrow \ |f(x)-L_2|<\varepsilon$.

But $|L_1-L_2|=|(L_1-f(x))-(L_2-f(x))|$, then by the triangle inequality:

$|L_1-L_2| \leq |L_1-f(x)|+|L_2-f(x)| < 2 \varepsilon$.

Hence $|L_1-L_2|$ is less than any positive real number, so $L_1=L_2$.

RonL
• December 24th 2005, 02:37 PM
ThePerfectHacker
Thank you CaptainBlack, is that your own proof because if it is you used the triangle inequality nicely.
I also have a question how can I use Equation Editor like you did.
• December 24th 2005, 04:27 PM
Jameson
It's called Latex and you type in these tags to use it on this site: [ math ] code here [ /math ] (but without the spaces). There's a tutorial on how to use it somewhere on this site, I'll do some searching.

Jameson
• December 24th 2005, 11:41 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
Thank you CaptainBlack, is that your own proof because if it is you used the triangle inequality nicely.
I also have a question how can I use Equation Editor like you did.

Is it my own proof: yes and no, the ideas are recycled from things I have
see probably including proofs of this result.

Equation formatting uses this sites built in LaTeX facilties, see:

RonL
• December 25th 2005, 12:33 PM
ThePerfectHacker
Thank you, the tutorial is rather complicated. Time for me to master it. Is Latex also used in other sites?
Funny if mathemations would have used latex code for real math (not interent) how messy it would have been.
• December 25th 2005, 01:15 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
Thank you, the tutorial is rather complicated. Time for me to master it. Is Latex also used in other sites?
Funny if mathemations would have used latex code for real math (not interent) how messy it would have been.

The LaTeX used here is a version of what is the de facto standard for writing
mathematical text. Most papers are produced in some version of TeX.

RonL