Here we list algebra of expressions that do not have a limit (not even
infinity or minus infinity). We will use the symbol *N* for a sequence
without a limit. In a Note at the end we will also briefly discuss some
indeterminate expressions involving *N*.

Since sequences whose limits do not exist are not encountered very often, there is usually little attention paid to this kind of algebra; in fact, the following list is probably the only one that you will find anywhere. This shows that this topic is not all that important, which is a good excuse for not listing all possible combinations; after all, sequences without limit can have all kinds of troubles in them and a complete list would be very long (especially if we wanted to treat indeterminate expressions properly).

**Addition/subtraction:**

*N* ± *L* = *N**L*.

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

**Multiplication/division:**

*N*⋅*L* = *N*,*N*/*L* = *N*,*L*/*N* = *N**L*.

Indeterminate expressions:
*N*⋅0,*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 some
expressions as indeterminate. For *N* we will use
^{n}

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

**The expression**
*N* ± ∞:

**The expression**
*N*:

**The expression**
*N*⋅*N*:

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

**The expression**
*N*:

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

**The expression**
*N*⋅∞:

**The expression**
*N*/∞:

**The expression**
*N*:

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

**The expression**
^{N}:

**The expression**
^{N}:

**The expression**
*N*^{∞}: