what is the absolute value of infinity? is it just zero?

it is undefined.

But |x| = x if x >= 0

The absolute value of infinity is not actually a defined value, as infinity is not a real number. However, if we think of it in terms of limits, we can say that the limit of the absolute value of a number as it approaches infinity is indeed infinity.

To understand this concept, we can use the definition of the absolute value. The absolute value of a number x is defined as the distance of x from zero on the number line. When we say that the absolute value of a number is infinity, we mean that the number is infinitely far away from zero.

Mathematically, we can express this by the notation:

|∞| = ∞

This means that infinity itself cannot be measured or quantified like real numbers can. It does not have a specific magnitude, and hence, there is no finite value that can represent the absolute value of infinity.