# Zero division

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1. Aug 9, 2015

### Stephanus

Dear PF Forum,
These rules are not int practice. Is this true?
Because 30 years ago in high school, I was taught.
x/zero is infinity, error in computer.
zero/zero is undefined, not zero. Also error in computer.

So zero/zero is undefined or zero?
Thanks for anyhelp confirming my doubt.
[Add: is not in practice today, implying that in the old days some calculation still use zero/zero = zero.
When was it's not in practice?]

2. Aug 9, 2015

### micromass

Staff Emeritus
This is wrong, something divided by 0 is undefined. Always.

Undefined.

3. Aug 9, 2015

### Stephanus

A. $\lim h{\to 0} \text{ } \frac{6}{h} \rightarrow \text{ infinity}$

B. $h = 0; \frac{6}{h} \rightarrow \text{ undefined}$

C. $\lim h{\to 0} \text{ } \frac{6h}{h} \rightarrow 6$

D. $h = 0; \frac{6h}{h} \rightarrow \text{ undefined}$
Could you please tell me which ones are wrong?
Thanks.

4. Aug 9, 2015

### Staff: Mentor

No.
The limit doesn't exist. If you don't understand why, read up on left-hand and right-hand limits.
Correct
Correct, although it would normally be written as $\lim_{h \to 0} \frac{6h}{3h} = 0$. In other words, without the last arrow, which indicates "approaches."

Edit: I miswrote 0 instead of 2 as the limit. It should be $\lim_{h \to 0} \frac{6h}{3h} = 2$
Yes

Last edited: Aug 9, 2015
5. Aug 9, 2015

### Stephanus

You mean $\lim_{h \to 0} \frac{6h}{3h} = 2$?
The "last arrow" I think is ambiguous in math.
I should have said $\lim_{h \to 0} \frac{6h}{3h}$ is equal/match to 2.

6. Aug 9, 2015

### micromass

Staff Emeritus
I want to be a bit more careful than what Mark says. Although in principle I agree with him, you seem to use arrows $\rightarrow$ everywhere, and it is not clear to me why you write this arrow. So the following are correct:

A. $\lim_{h\rightarrow 0} \frac{6}{h}$ is undefined.
B. If $h=0$, then $\frac{6}{h}$ is undefined.
C. $\lim_{h\rightarrow 0}\frac{6h}{h} = 6$.
D. If $h=0$, then $\frac{6h}{h}$ is undefined.

7. Aug 9, 2015

### micromass

Staff Emeritus
Yes.

It's not ambiguous, it's just wrong to use it in that context.

8. Aug 9, 2015

### Stephanus

I should have asked directly to Mark44, but your post is the last one.
Actually this
doesn't have anything to do with "left-hand and right-hand limits", because it's the last arrow that I put wrongly? I just read left-hand, right-hand arrow, http://www.millersville.edu/~bikenaga/calculus/limlr/limlr.html [Broken]
but it doesn't explain why there's no limit.

Last edited by a moderator: May 7, 2017
9. Aug 9, 2015

### micromass

Staff Emeritus
The point of the left-hand and right-hand limits is the following:
While
$$\lim_{h\rightarrow 0} \frac{6}{h}$$
is undefined. It is true that the left-hand limit
$$\lim_{h\rightarrow 0-} \frac{6}{h} = -\infty$$
and the right-hand limit is
$$\lim_{h\rightarrow 0+}\frac{6}{h} = +\infty$$

10. Aug 9, 2015

### Stephanus

Ahh, so if the left hand and the right hand lmits differs very big, than we can say "There is no limit"?
[Add: differ is the wrong word I think. So, the left-hand and the right-hand limit must match/exact?]
[Add: $\lim_{h \to 0+} \frac{6h}{h}$ is equal to $\lim_{h \to 0-} \frac{6h}{h}$, so we can say that there is a "limit" there?]

11. Aug 9, 2015

### micromass

Staff Emeritus
Yes. A limit exists if and only if the left-hand and right-hand limits exist and are equal.

12. Aug 9, 2015

### Stephanus

Thanks! A new concept. This helps me much in understanding math (at least about this "limit" thing . Left hand should match right hand)

13. Aug 9, 2015

### HallsofIvy

Staff Emeritus
Part of the problem is that you are trying to interpret "infinity" as if it were a number- it isn't. Saying that a limit "is equal to infinity" is the the same as saying the limit "does not exist", just in a particular way.

14. Aug 9, 2015

### Staff: Mentor

Yes, that's what I should have written. I have edited my earlier post to indicate this.