Zero/Zero, infinity/infinity = ?
- Thread starter Saint
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They aren't. They're undefined.
You still haven't learned your prior lesson, have you? Is $$\frac{sinx}{x}$$ when $$x->0$$ undefined?
The precise answer is given by a well-known calculus theorem.
As is, they are undefined. If it is a result of a limiting process (like sinx/x as x -> 0), then there are procedures (L'Hopital's rule) to handle it.
Infinity is not a member of the Reals and therefore it is meaningless to try to do algebra with it unless you define the relevant rules which extend the Reals appropriately. Similarly 0/0 is meaningless because it represents 0 * (1/0), ie when you divide by X you actually multiple by the number which when multiplied by X gives 1. There is no Y such that Y*0 = 1 in the Reals and therefore 1/0 is no more a valid expression than elephant/cheese = table.
Careful, Saint is easily confused.
What eram said true: zero/zero is undefined.
$$\lim_{x\to 0}\frac{\sin x}{x} = 1$$
$$\frac{\sin 0}{0}$$ is undefined.
$$\lim_{x\to 0}\frac{\sin x}{x}$$ is not the same as $$\frac{\sin 0}{0}$$.
Exactly, which is why if you define a function $$f(x) = \frac{\sin x}{x}$$ you have to define the value of f(0) separately. If you decide f(0) = 1 then that fits with the limit but that isn't required.