# Thread: When does a limit not exist?

1. ## When does a limit not exist?

Hi

Kinda of a fundamental question here.

When does one say that a limit exists or not exist?

For example)

$\displaystyle\lim_{x\to 0^{+}} \frac{1}{x} = \infty$

This limit exists and is equal to infinity?

Is a typical case of a limit NOT existing $\displaystyle\lim_{x\to 0} \frac{|x|}{x}$ ? Because the limit is either negative one or plus one depending on if you approach zero from left or right.

How can we really say that a limit 'equals' infinity? Infinity is not a number.

Thx

2. Originally Posted by Twig
When does one say that a limit exists or not exist?
For example) $\displaystyle\lim_{x\to 0^{+}} \frac{1}{x} = \infty$
This limit exists and is equal to infinity?
Is a typical case of a limit NOT existing $\displaystyle\lim_{x\to 0} \frac{|x|}{x}$ ? Because the limit is either negative one or plus one depending on if you approach zero from left or right.
How can we really say that a limit 'equals' infinity? Infinity is not a number.
We can start a real argument right here and now with this question.
The are many authors who write calculus textbooks that say
$\displaystyle\lim_{x\to 0^{+}} \frac{1}{x}$ does not exist.
Some say the expression increases without bound.
Still others have a vague way of extending the real numbers to include unbounded entities for which the symbol $\pm\infty$ is used.

3. I see =)

Yes I agree. I see many people saying that the limit equals infinity.
My calculus teacher hated that.

Thx

4. Originally Posted by Twig
I see =)

Yes I agree. I see many people saying that the limit equals infinity.
My calculus teacher hated that.

Thx
Good for your teacher. I would say "the limit does not exist and is infinity!" Because saying a limit is "infinity" is just shorthand for saying the limit does not exist in particular way. Many Calculus texts will say that "1/x, as x goes to 2, converges to 1/2" but that "1/x, as x goes to 0, from above, diverges to infinity."

5. Well, this is what I have read and believe it to be true.
$\lim_{x\to 0}\frac{1}{x^2}=\infty.$
The limit is said to exixt infinitely

6. Originally Posted by pankaj
Well, this is what I have read and believe it to be true.
$\lim_{x\to 0}\frac{1}{x^2}=\infty.$
The limit is said to exixt infinitely
This limit does not exist because there is no value of epsilon that can be chosen such that

$|f(x)-L|<\epsilon$ whenever $|x-a|<\delta$

I mean, how would you even approach the proof?

$|f(x)-\infty?| whenever $|x-0|<\delta$

When we write $\lim_{x\to{a}}f(x)=\infty$, we are saying that the limit does not exist, but we are also saying why it doesn't exist.

7. Originally Posted by Twig

Is a typical case of a limit NOT existing $\displaystyle\lim_{x\to 0} \frac{|x|}{x}$ ? Because the limit is either negative one or plus one depending on if you approach zero from left or right.
Thx
Yes. This limit DNE because $\lim_{x\to{0^-}}\frac{|x|}{x}\neq\lim_{x\to{0^+}}\frac{|x|}{x}$.

8. ## Argueable

I've always been taught that infinity is a concept, not a number. It just allows you to understand the long term behavior of a function. up to you