The statement 1/infinity=0 is the informal, though mathematically inaccurate form of the following statement:
                  lim 1/x = 0         (1)
                  x→∞
Or in words: the limit of 1/x, as x approaches infinity equals zero. As x gets (arbitrarily) large, 1/x gets (arbitrarily) small.

Note that x is never equal to infinity, but x is unbounded for infinitely increasing values. Infinity is not a number but a mathematical concept. Numerical operations such as multiplication, division, addition and subtraction are not applicable to infinity, because this results in paradoxical constructs, such as ∞ + 1 = ∞, ∞ - 1 = ∞, and 2 × ∞ = ∞. To understand why these definitions are false, consider for instance the class of natural numbers, and the class of even numbers:

natural numbers: 1, 2, 3, 4, 5, …
even numbers   : 2, 4, 6, 8, 10, …

Both classes, the natural numbers and even numbers are unlimited classes, as they contain infinitely many elements. However, any natural number can be multiplied by two, to form an even number. There is a one-to-one correspondence between the classes of the natural numbers and even numbers. Multiplying the natural numbers by a factor two, halves the number of elements in the dataset, but the total number of elements in the set is still infinite. Thus, numerical operations do not apply to infinity, and infinity itself cannot be treated as a number.

A second point that frequently arises is whether the limit of x→∞ of 1/x is equal to zero, or approaches zero. Mathematically:

                  lim 1/x = 0
                  x→∞
or
                  lim 1/x → 0     (?)
                  x→∞
The former statement is correct, and this follows from the definition of the unbounded limit:
                  lim f(x) = y0    (2)
                  x→∞
This definition means that for every M>0 (large), there is an ε>0 (small) so that x>M implies |f(x) - y0|<ε

The proof of (1) now follows from the definition of (2). Since y0=0, we have:

                  |(1/x)-0| = |1/x| < ε

                ↔ -ε < 1/x < ε

                ↔ 1/ε < x

Thus, if we set M=1/ε, then for all x with x>M:

                  |1/x| < ε
Which means that:
                  lim 1/x = 0         (1)
                  x→∞

And finally, the statement 1/∞=0 is an informal statement that is commonly used, even by some mathematicians. There is no real problem with this statement in daily usage, as long as you keep in mind that the statement implies a limit of 1/x for x approaching infinity. However, if you are doing a math test, I suggest you use the proper definition.

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