Division by Zero.....

2+2=5 is a song.....by radiohead.

it doesnt involved any equation...right?

 

The limit problem I have presented is supposed to illustrate one example why anything divided by zero is undefined. The key concept to know is what the term "limit" means.

The mathematical definition of limit is the value of which the given function approaches to as the given input, (usually X), either approaches BOTH SIDES of a certain point of interest (in this case, it's 0), or approaches infinity.

With that definition out of the way, let's take a look at an example:
Find the limit of the function f(x)=1/x as x approaches 1. I'll borrow Marcy's graph since I'm lazy.



The point of interest as well as the vertical line where x=1 has been indicated in red on the graph. We are interested in the limit as x approaches 1 from BOTH SIDES of that point, so we trace the graph from both sides of x=1.


(Yeah, shitty Paint job, but oh well...)

If you were to trace the graph of both sides where x=1, you will see that both sides meet at the same point, in which the function (or y-coordinate) happens to equal 1. So in this example, the limit of the function, f(x)= 1/x as x approaches 1 is 1. Which makes sense because after all, 1 divided by 1 equals 1.

Now, let's go back to the original problem which is to find the limit as x approaches 0 (which, at x = 0, it would be where we are dividing by zero in the given function, f(x) = 1 / x ). We will go ahead and trace the graph from both sides where x = 0.



Now, look what happened here. As we trace both sides where x = 0, both of our traces do not converge at all. So in this case, the limit is undefined, or 'does not exist'.

Hence, anytime you take any given number or term and divide it by zero, the result is undefined.

Read above. Also, for your information, the math and scientific community has formally accepted that anything divided by zero is undefined due to mathematical reasons.

The IEEE standard design of the floating point unit forced any number (other than 0) divided by 0 to be "infinity", and 0/0 to be "NaN" (not-a-number). The former is used for certain applications in which it is allowed given that you are working with positive numbers only, for if you take the one-sided limit as x approaches 0 from the positive side, the result is infinity. However, in regards of computer software, dividing by zero does cause programs to crash as "runtime errors". So no, computers don't have to handle divide-by-zero cases at all except by crashing as Loonylion mentioned.

 

The limit of 1/x as x approaches zero is like the limit of my love for Hypr as my thoughts approach the taboo of male love: undefined.

 

I think f(x) = 1/x is the kind of function you call partially continuous so it could have a partial limit too.
But I agree with you it has no single limit and no two. It's divergent, from the left and from the right.

as x -> 0+0 f(x) -> +infinity (from the right)
as x -> 0-0 f(x) -> -infinity (from the left)

Yes, that's no limit, but a asymptote. (and I never used the word limit in my previous post)

Not my graph. I stole it, too.

EDIT:
I just remembered this nice function.
f(x) = sin (1/x)

 

stop torturing my brain! xD

 

I divide by Zero every month.....to workout my budget

 

There is no such thing as a "partially continuous" function. Either the function is continuous, or discontinuous. In this case, the function, f(x) = 1/x is discontinuous at x = 0.

Also, you have misused the term "partial limit". Partial limits are only applicable for functions of multiple variables.

 

Umm... yes sorry, I tried to point out a little error in you definition.
(It's just really hard to properly translate highly specialized mathematical expressions)

With "partially continuous function" I meant a function that is discontinuous, but where all the discontinuous points are borders for the continuous intervals of that function. (seems to be very uncommon in use)

Anyways, the actual definition of a limit is this.

in simple words: if x gets very close to a we get L as the limit

The restriction that a limit must exist for both sides can't be found in this definition. (neither explicit nor implicit)
But still, we say f has a limit at x0 if the limits from both sides at x0 are equal.
(that doesn't mean you hide their existence if they aren't equal)

limx→0 1/x doesn't exist, but the one-sided limits do, and if you get the task to calculate limx→0 1/x it's not the best thing to just sit there and say 'it's undefined'.
It's way better to give the one-sided limits. (since they exist)
limx→0+0 1/x = +∞
limx→0-0 1/x = -∞
(or in short: limx→0±0 1/x = ±∞)

There are many sources agreeing with limx→0±0 1/x = ±∞:
http://en.wikipedia.org/wiki/Table_of_limits
http://wwwx.cs.unc.edu/~gb/wp/wp-content/uploads/2007/01/democalculus.pdf
http://www.slideshare.net/leingang/lesson-3-the-limit-of-a-function-presentation?src=related_normal&rel=253377
http://www.slideshare.net/leingang/lesson-6-continuity-ii-infinite-limits
[quote author=http://calculus7.com/id1.html]The following 2 animations deal with limx->0 sin(1/x) and limx->0 xsin(1/x).

In the first case as we zoom near 0 sin(1/x) oscillates at an increasingly rapid rate between -1 and 1 and hence approaches no limit. The reason is as x → 0 1/x→∞ and sin(x) does not approach a limit at ∞.[/quote]

Moreover when we're extending the real numbers to a real projective line 1/0 = ∞ is true, even for normal calculations.

NOTE: I know that ∞ can hardly be called a limit, since ∞ is no number, but limx->x[sub]0[/sub] f(x) = ∞ is a common syntax to express that the function is divergent at x0.

 

Me+apples=none for anyone that's for sure XD


...all this over a zero-man I wish I had more math knoledge than year 10

 

this topic make me feel like i just open a can of worms >.>

anyway thanks Hypr for trying to explain......