Why does Michio Kaku say that $\frac{1}{0} = \infty$? Why does Michio Kaku say that $\frac{1}{0} = \infty$?
http://youtu.be/AJ4zlvqOtE8?t=4m43s
Instead of $\frac{1}{0}$ that's not defined, so we don't know.

This is context-dependent.  For some purposes, in particular in projective geometry, in trigonometry, in dealing with rational funtions, it makes sense to have a single object called $\infty$ that's at both ends of the real line, so that the line is topologically a circle.  In other contexts it makes sense to distinguish between two objects, $\pm\infty$.  Any of these three things can in some instances be the limit of a function.
I don't agree with his statement that to mathematicians, infinity is simply a number without limit.  A variety of different concepts of infinity exist in mathematics.  There are some things that must be considered infinite numbers, including (1) cardinalities of infinite sets and (2) infinite nonstandard real numbers and (3) some other things.  (1) and (2) in this list are definitely not the same thing.  There are also the infinities involved in things like the Dirac delta function $\delta$, where, loosely speaking, one says $\delta(0)=\infty$, but notice that $2.3\delta$ is different from $\delta$, so this "$\infty$" is not "simply a number . . . . . .".  There is the $\infty$ of measure theory, satisfying the identity $0\cdot\infty=0$, and thre are the $\infty$s of calculus, in which $0\cdot\infty$ is a indeterminate form.  This is far from a complete enumeration . . . . . .
