An equation that uses Divide by zero might go to "max limit" or to zero (in practical terms), based on preceding values. It will not suddenly say; "Sorry, the resultant value is indeterminable".
Using an function where its undefined is fundamentally invalid.
lets say I define function:
f(x,y) = 1+ x/y; for all y in R such that y>0; for all x in R such that x>=0
If I call this function f(-4,2) same thing. Its not defined, its NOT defined for xbased on preceding values
Its not hard to completely break this.
Consider sin(1/x)
http://www.wolframalpha.com/in...
See what that does? It doesn't merely shoot off to infinity; its clearly bounded between 1 and -1. But nor does it have a clear value at x=0; its limit is undefined at 0.
http://www.wolframalpha.com/in...
Limit does not exist.
It does go towards a finite number, it doesn't go to plus or minus infinity. Its JUST not defined. Don't ask for a value there because there isn't one is the ONLY correct answer.
The point is, a function is defined on an interval. If there is somewhere where its undefined, it is an error to ask it for a value there. If you are interested in where the function is going, then don't ask it for a value, as it where it going (ie take a limit). But don't be too surprised when you find out there isn't a limit either.
And don't forget that lots of real world functions have values that don't line up with their limits.
f(x) = { 2x for all x in R, such that x != 5; 5 if x = 5 } this is a perfectly valid function. The limit of f(x) as x->5 = 10 from the left, and the right; but f(5) = 5.
You are technically correct but IMHO practically wrong. I'm talking about "real use" such as in animation, graphing and financial.
I can't imagine a scenario in financial where division by zero should ever be fudged.
I can't imagine a scenario in graphing where division by zero should ever be fudged.
In amimation; yes; sometimes you take shortcuts to correctness for performance, but honestly handling division by zero properly tends to be an optimization. You are usually dealing with a vertical line (infinite slope), or a the intersection of a line parallel to a plane or something; and you can and should handle it correctly easily.
Can you cite an example where not handling a case where division by zero is going to occur properly makes sense to you? Rather than just hand waving that examples exist, can you acutally provide one? Because I can't think of one.