Why 0 ÷ 0 = 1

Great Debates
41

Ahhh, feathermuck.

We can figure out what the function f(x) = (x[sup]3[/sup]-8)/(x-2) ‘should’ be from the limit, should we want to define it at x=2: since the limit is 12, defining f(2)=12 turns f into a function that is continuous at 2.

It helps that this particular function has a nice, well-defined limit there.

OTOH, when we’re talking about the entire class of indeterminate forms whose numerator and denominator both approach 0 as x gets near to some value, such well-definedness goes right out the window. Each such expression has its own limit (if it has a limit at all), and the values of these limits are all over the place. So as others have pointed out here, there is no single value that 0/0 ‘ought’ to be, hence calculus is no help, except in a negative sort of way - reaffirming our intuition that there’s no appropriate definition for 0/0.

42

Yes, I know what a limit is. But, the point is that you are considering what is happening arbitrarily close to a point which gives one an idea of what one might want to define as the value at the actual point…if one wants to make the function continuous at that point, for example (as RTFirefly points out).

I still disagree. I think we learn from limits an important negative conclusion, namely that it doesn’t really make sense to define 0/0 to be 0 or 1 or anything like that since the limit of the ratio of continuous functions that both approach 0 can in fact be any value.

I agree with what RTFirefly has said except as a matter of connotation because I think this “negative sort of way” in which calculus is of help is important enough that I would not say that “calculus is of no help, except…”. I think it has helped us to see another reason why it is most sensible to consider 0/0 to be indeterminate and this is important enough to phrase in a more positive light like “Calculus is of help here because…” But, this issue of connotation is at some point getting down to philosophy and interpretation more than mathematics.

43

Using WindowsNT calculator program, calc.exe. It gives me this when dividing 0/0: Cannot divide by zero. Or maybe I need a Windows Update. :smiley:

44

This one got a limit. there’s a factor (x-2) that can be taken out.

45

RT mentioned that. Go back and read his post again.

46

lim sin(x)
[sup]x-›0 X[/sup]
= 1

Funny, this lim(sinx/x) =1 thing is discovered by experiment rather than by substitution. If using substitution, it would be no limit, as initially thought.

47

I didn’t see his post.

48

You are misusing terminology here. The fact that you cannot find out what sin(x)/x is when x=0 by substitution does not mean that lim(sin(x)/x) is not well-defined. And, you don’t have to get the limit “by experiment”; you can get it using calculus techniques (for example, either by l’Hopital’s rule or the power series expansion for sin).

49

Hear, hear.

Which one was the first - the known result of Lim sinx/x and the known name of l’Hopital?

50

Everyone send me dollars to my PayPal account. I’ll send you nothing in return. :smiley:

51

Look, I know this is just noodling, but I’m still wondering about the limit of x/x as x -> 0. The other limits folks have mentioned are all going to 0/0, but the numerator and denominator are approaching 0 at different rates.

52

Rates don’t enter into the definition of a limit at all, so that’s pretty much out.

53

lim (x/x) is 1 as you can see by plugging it into your calculator and making x smaller and smaller. In fact, x/x is identically 1 for all values of x except 0.

54

You all are so funny. Math has no relationship to reality, you sillies.
Many of you all amuse yourselves with assumptional vanities.
Bless you, sweet innocent hearts, what a funny. funny joke.

55

Is the irony of your posting this on the internet using a computer lost on you?

56

So what is the correct answer for lim (x->0) x/x?

It is clearly 1 “by inspection”, but is this what proper calculus methods say?

Would someone care to do l’Hopital’s rule on this?

57

Although I find the various explanations and arguments on this topic somewhat interesting, I have to laugh to myself on how they have presented their arguments in such an oversimplified and primitive fashion. These posts by these amateur mathematicians leave much to be desired and don‘t even come close to fully illuminating the concept .

If I may explain and make everything clear:

My hero, zero, you’re such a funny little hero but till you came along, we counted on our fingers and toes! Now you’re here to stay and nobody really knows how wonderful you are, how you could never reach a star, without you zero, my hero, how wonderful you are, how wonderful you are!

you place a zero after one, and you’ve got yourself a ten - see how easy that is, when you run out of digits you can start all over again - see how simple that is!

that’s why with only ten digits, including zero, you can count as high as you could ever go, forever, towards infinity - noone ever gets there, but you could try!

with ten billions zeros, from the cavemen who invented you, they counted on their fingers and toes (and maybe some sticks and stones, rocks & bones, neighbors toes) oh yeah, nobody really knows how wonderful you are, how you could never reach a star, without you zero, my hero, how wonderful you are, how wonderful you are!

you place a zero after any number and you’ve multiplied that number by ten - see how convenient that is, you place two zeros after any number and you’ve multiplied that number by one hundred - see how important that is etc, etc, ad infinitum, ad astra, forever & ever, with zero, my hero, how wonderful you are

58

You don’t need l’Hopital’s rule for this, because it’s equal to 1 whenever x is non-zero. That means the limit is 1, cause the function is constant as you approach zero.

59

Would the zero-point be represented as an open circle if it were graphed?

60

Yep.

btw, it’s pretty easy to show that if a function is equal to a constant c on an interval surrounding a point p, then the limit as x goes to p of f is c. That’s the principle at work here, not any kind of inspection.