Question

If something is divided by zero, then it is defined as +-infinity, rather than being undefined, is this line of thinking correct? Why or why not?

Answer 1

The traditional answer is no, it's just undefined. However, if you consider the fact that division gives monotonically larger results as you divide by smaller and smaller (in terms of magnitude, disregarding sign) numbers, you could say that the limit as x goes to zero of 1/x is infinity, and by graphing you can see that it's actually negative infinity on the negative side of zero and positive infinity on the positive side.

Answer 2

More specifically, 'the limit' goes to infinity. Just by itself, it would be considered undefined (or indeterminate) not sure which one is proper in this case.

Answer 3

You should take into account that +- inf are on the extended real number line, they are not themselves real numbers (the real line).

Getting a limit for p/0 is not a clear cut exercise, it just depends.

Answer 4

hold the phone. as betwixt said, as a limit statement the limit does not exist, since from one side it is positive and the other negative infinity. so please know what
\[\frac{a}{0}\] is simply not a number.

Answer 5

there is no such thing as infinity

Answer 6

done

Answer 7

think of it this way: if i drive 100 miles on 10 gallons of gas i get
\[\frac{100}{10}=10\] miles to the gallon and if i drive 100 miles on 5 gallons of gas i get
\[\frac{100}{5}=20\] miles to the gallon and if i drive 100 miles on 2 gallons of gas i get
\[\frac{100}{2}=50\] miles to the gallon, but if i bike 100 mile the question "what is your gas milage?" makes no sense

Answer 8

Well, I agree that there is only one of them (infinity) :-)

Answer 9

you cant say a number equals infinity , its just that people continue to abuse notation etc and say that anything over zero is "infinity" , its not defined, end of story

Answer 10

elecegineer is completely correct, so please please don't let anyone tell you that something over 0 is infinity. it is not true, and not even true as a statement about limits

Answer 11

There's a pretty serious distinction between the statement that no number equals infinity, which is fairly obvious, and the statement that "there is no such thing as infinity." Yes, the limit is undefined in this case since the function doesn't approach the same value from both sides, but that doesn't mean there's never a conceptually valid time to talk about infinity...

Answer 12

For example, the function 1/x2 can be made continuous (under some definitions of continuity) by setting the value to +∞ for x = 0, and 0 for x = +∞ and x = −∞. The function 1/x can not be made continuous because the function approaches −∞ as x approaches 0 from below, and +∞ as x approaches 0 from above.

http://en.wikipedia.org/wiki/Extended_real_line

Answer 13

Stir, stir..

Answer 14

Nice discussion, but it may not be helpful to a guy taking Calculus I, a course taught by a guy with a PhD who says the lim of a/0=infinity

Answer 15

I take your point although I don't see it as being any more or less helpful than asserting the existence of an infinite set in the first instance, it's all a matter of axioms and definitions and the student might as well get used to that idea.