What is infinity times the reciprocal of infinity? I was talking with a friend about interesting properties of numbers and their theoretical contradictions and solutions when we came up with this. What is the answer?
So...
$x * ∞ = ∞$
and...
$\frac{1}{x}*x=1$  
So what do you get when you do...
$\frac{1}{∞}*∞$? $∞$ or $1$?
 A: John Wallis, who originally introduced the symbol "$\infty$" in the 17th century, used it to denote a specific infinite number, and furthermore exploited an infinitesimal of the form $1/\infty$ in area calculations. Extended number systems that include infinitesimals were used by such giants as Leibniz, Euler, and Cauchy, and are in active use today.  
Today the symbol $\infty$ is not generally used in this sense.  However, the wording of the question by the OP suggests that this is the sense he may have in mind, rather than the traditional meaning in the context of "indeterminate forms".  The OP should keep in mind that the modern use of the symbol is different from its historical use.
If $H$ is an infinite number (meaning that it is greater than every real number), then $H \times \frac{1}{H}=1$ as for ordinary real numbers.  See for example http://www.math.wisc.edu/~keisler/calc.html
The answer given by user mixedmath assumes that you are dealing with "indefinite forms", and is also correct.
A: Unfortunately, infinity is not a number, and cannot be manipulated as such.
For example, $\lim_{n \to \infty} x = \infty$, and $\lim_{n \to \infty} x^2 = \infty$, but 
$\lim_{n \to \infty} \frac{x}{x} = \frac{\infty}{\infty} = 1$,
$\lim_{n \to \infty} \frac{x}{x^2} = \frac{\infty}{\infty} = 0$,
$\lim_{n \to \infty} \frac{x^2}{x} = \frac{\infty}{\infty} = \infty$,
In fact, we can make $\frac{\infty}{\infty}$ converge to anything we want.
