What follows is a list of algebra expressions that can be used for calculating limits. It is an extension of the usual algebra. We also mention indeterminate expressions here. They are dealt with in more detail here. Here you find a brief list of the limit algebra if you want just a summary.

All rules written here as limit algebra are supported by theorems proved in
textbooks, for instance the rules
*L* = ∞^{+} = ∞

If

fgoes to infinity ataandgconverges to a real numberLata, thengoes to infinity at f+ga.

Iffconverges to 0 ataandon some reduced neighborhood of f> 0a, then1/ goes to infinity atfa.

And now the rules.

**Addition/subtraction:**

Real numbers add/subtract in the usual way.

Infinity behaves nicely in most cases:

*L* = ∞*L*,

Indeterminate expression:

**Multiplication/division:**

Real numbers multiply in the usual way, also division works as usual as
long as the denominator is not equal to zero.

^{+} = ∞^{-} = −∞.

Infinity behaves nicely in the following cases:

*L* = ∞*L* = ∞*L*,

*L* = −∞*L* = −∞*L*,

*L*/∞ = 0*L*,

Indeterminate expressions:

The expressions *L*/0*L* and
*L*/0=*L*⋅1/0)^{+} and 0^{-}. If the zero is not of these two types,
that is, if it (during the limiting procedure) keeps changing sign, then the
limit

**Powers:**

Real numbers work in powers *A*^{B} in the usual way
for positive *A*. When *A* is not positive, one has to be cautious,
in particular 0^{0} is an indeterminate expression (see below).

Infinity behaves nicely in the following cases:

^{L} = ∞*L*,
^{L} = 0*L*,

*L*^{∞} = ∞*L* > 1,*L*^{∞} = 0*L*| < 1,*L*^{∞} DNE*L* < −1,

*L*^{−∞} = 1/*L*^{∞} = (1/*L*)^{∞},*L*^{−∞} = 0*L*| > 1,*L*^{−∞} = ∞*L* < 1,*L*^{−∞} DNE*L* < 0.

^{∞} = ∞.

Indeterminate expressions:
^{∞},^{0},^{0}.

**Important note:**
General powers have to be
handled in the basic *e*^{ln}

^{+})^{0} = *e*^{0⋅ln(0+)}
= *e*^{0⋅(−∞)}
= *e*^{-0⋅∞},

To be able to evaluate limits one also needs to know the values of elementary functions at endpoints of intervals of their domains.

**Dictionary:**

*e*^{∞} = ∞,*e*^{−∞} = 1/*e*^{∞} = 1/∞ = 0.

^{+}) = −∞.

*k*π)^{+}) = −∞,*k*π)^{-}) = ∞,*k*π)^{+}) = ∞,*k*π)^{-}) = −∞,

^{+}) = ∞,^{-}) = −∞.

**Note:** The basic difference between a "real" algebra and the limit
algebra is that here a number does not represent a fixed quantity, but the
outcome of some process. We can thus imagine that, say, 3 is actually "almost
3". This explains why some things do not work the way one would expect. For
instance, from the usual algebra we know that 1 to anything is 1. However,
the expression
^{∞}