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`) = y_{0} (2)
`x`→∞

This definition means that for every M>0 (large), there is an ε>0
(small) so that

`x`>M implies |f(

`x`) - y

_{0}|<ε

The proof of (1) now follows from the definition of (2). Since
y_{0}=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.