Here we list (a part of) algebra of expressions that do not have a limit (not
even infinity or minus infinity). We will use the symbol *N* for a
limit that does not exist. In a Note at the end we will also briefly discuss
some indeterminate expressions involving *N*. Some examples there are
quite involved and interesting, so we look at them in
Solved Problems.

Since this kind of considerations is done rarely, the necessary properties are usually worked out on the spot. We made a list here since we were curious and also for the sake of completeness, but since everybody seems to live happily without it, we took it as a good excuse for not listing all possible combinations (there is too many of them).

**Addition/subtraction:**

*N*±*L* = *N**L*.

Indeterminate expressions:
*N*±*N**N*±∞.

**Multiplication/division:**

*N*⋅*L* = *N**N*/*L* = *N**L*.

Indeterminate expressions:
*N*⋅0,*N*⋅*N*,*N*,*N*/*N*,*N*,*N*/0,*N*⋅∞,*N*/∞,*N*.

**Powers:**

*A*^{N} = *N**A* > 0,*A* not equal to 1.

Indeterminate expressions: for instance
*N*^{A},^{N},^{N},*N*^{∞}.

We will show some examples to convince you that we correctly listed those indeterminate expressions.

**The expression** *N* + *N*:

**The expression**
*N*±∞:

**The expression** 0⋅*N*:

**The expression** *N*⋅*N*:

**The expression** 1/*N*:

**The expression** *N*/*N*:

**The expression** 0/*N*:

**The expression** *N*/0:

**The expression**
*N*⋅∞:

**The expression**
*N*/∞:

**The expression**
∞/*N*:

**The expression** *N*^{A}:

**The expression** 1^{N}:

**The expression**
∞^{N}:

**The expression**
*N*^{∞}: