Isn't 1/(1/∞) just 1*(∞/1) = ∞?

The problem is that a limit is only a numerical quantity when it approaches a numerical quantity. If a limit approaches infinity, that's not the case.

There are certain ways that ∞ can be treated, in the correct circumstances, as though it were numerical. Those ways are restricted because if we violate the restrictions, the result is nonsense. And it's nonsense to attempt to define an answer in the case of division by zero.

No.

The limit as x approaches infinity of 1/x is zero. In this case, you're dividing a finite number by a finite number which is increasing without bound, and you get a result that is not equal to zero, but approaching zero.

1/0 = infinity is false. In that case, you are trying to divide by zero, which is not allowed in math, and you are using infinity as a number, which it's not. A number can "approach infinity" meaning that it's getting bigger and bigger forever, but it cannot "equal infinity", which is meaningless.

You are treating ∞ as a numeric digit which you shouldn't.

You may say:

lim x-->∞ 1 / (1/x) = lim x-->∞ x = ∞

Yes, you're right. The answer is infinity.

Can anyone answer my math questions?

You have to use the arithmetic of limits to manipulate expressions like this. There are rules for expressions that evaluate to inf, -inf, or 0, etc.

There are also indeterminate forms which have to be handled differently.

1 / (1/x) = 1 * (x/1) = x

Oh I see what you did there.... 1/infinity = 0 so your.... witty

Yes it does