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. Be sure to look at Notes at the end for proper understanding of the rules, especially at the very last note. Here you find a brief list of the limit algebra if you want just a summary.

**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.

^{+} = ∞^{-} = −∞,^{+} stands for a sequence tending to
zero whose terms are all positive and 0^{-} stands for a sequence
tending to zero whose terms are all negative (in fact, it is enough to have
the positivity/negativity valid only at the "end" of the given sequence, that
is, after dropping its "beginning", a finite number of terms). See the note
at the end.

Infinity behaves nicely in the following cases:

*L* = ∞*L* = ∞*L*,

*L* = −∞*L* = −∞*L*,

*L*/∞ = 0*L*,

Indeterminate expressions:

The expressions *L*/0*L* not zero and

**Powers:**

Real numbers work in powers in the usual way; in particular, the power
*A*^{B} works as usual 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}.

**Some elementary functions:**

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

^{+}) = −∞,^{+} stands for a sequence tending to zero whose
terms are all positive (or at least they are positive if the first few are
dropped).

In the above equalities any given symbol does not stand for an actual number;
rather, it symbolizes sequences whose limit is equal to this specific symbol.
For instance,

One to infinity is in fact a shortcut for a sequence that converges to 1
raised to a sequence that converges to infinity. Since we perform the
operation term by term, we obtain the sequence of powers
*a*_{n} ^{bn}},*a*_{n} are eventually close to 1, but
they need not be exactly one. For instance, they might all be slightly more
than one. Since they are raised to huge numbers (the exponents
*b*_{n} tend to infinity), they may yield big answers and
the resulting sequence may even converge to infinity. It all depends on
balance. If the exponents become really big before the bases have a chance to
get close to 1, then "one to infinity" may be a big number. On the other
hand, if the bases get really close to 1 really fast, while exponents are
growing to infinity slowly, then the closeness of bases to 1 may outweight
the growth of exponents and "one to infinity" could be close to one.

Similarly, if the bases are less than one and tend to one slowly, while
exponents get big really fast, then "one to infinity" may be small, even
zero.

Similar balance decides the outcome of other indeterminate expressions, for
instance the outcome of

It is enough to investigate *L*/0

Recall that "0" stands for some sequence
*a*_{n}}*a*_{n}. If all *a*_{n} are positive
(this is denoted by 0^{+} in the limit algebra, it is actually enough
if the terms become positive after the first few are dropped), it follows
that we always divide "almost one" (recall that "1" stands for some sequence
*b*_{n} tending to 1) by a small positive number,
obtaining a huge positive answer. Therefore the resulting limit should be
infinity. On the other hand, if all *a*_{n} are negative
(this is denoted by 0^{-} in the limit algebra, it is actually enough
if the terms become negative after the first few are dropped), it follows
that we always divide "almost one" by a tiny negative number, obtaining a
negative number that in absolute value is very huge. Therefore the resulting
limit should be negative infinity. In this way we obtain the two formulas for
^{+} = ∞^{-} = −∞.

The third alternative is that as we follow the sequence
*a*_{n} towards its end, no sign prevails; that is, no
matter how far we go along the sequence, there are always negative and
positive *a*_{n} further on. This can be also expressed
like this: It is not possible to make the sequence have the same sign by
ignoring its beginning, that is, by dropping a finite number of its terms.
Since the sequence *a*_{n} tends to zero and
*b*_{n} is about one for large *n*, the numbers
*b*_{n} /*a*_{n}*b*_{n} /*a*_{n}*b*_{n} /*a*_{n}

The conclusion is that when we see the type

For some examples of this behaviour, see indeterminate expressions.

In this note we put together the two ideas from the previous notes to explain
why sometimes we need to consider one-sided limits.

The fact that a number in limit algebra is not exactly this number, but a
symbol for a convergent sequence, allows us to do things which could not be
done with numbers. For instance, we know that we cannot substitute the
number 0 to the function *x*^{2}.^{2} = ∞*a*_{n}}*a*_{n} are different from zero, then
*a*_{n}^{2}*a*_{n}^{2}^{2} = 0^{+}

Sometimes when we try to substitute a number representing a convergent
sequence into a function, we have to be careful and do extra work. For
instance, what is the outcome of *a*_{n}}*a*_{n} > 0.*a*_{n})

We marked the sequence on the real line, the values of
*a*_{n})

We had a notation for such "0", in the previous note we called it
0^{+}, and we just argued that
^{+}) = −∞.

These considerations are used quite often and deserve a formal definition.

Definition(one-sided limits)

Let{ be a sequence that converges to a real numbera_{n}}A.

Iffor all a_{n}>An, we say that{ converges toa_{n}}Afrom the right. We denote itand in the limit algebra we would use the symbol a_{n}→A^{+},A^{+}for such a sequence.

Iffor all a_{n}<An, we say that{ converges toa_{n}}Afrom the left. We denote itand in the limit algebra we would use the symbol a_{n}→A^{-},A^{-}for such a sequence.

In the following pictures we show how one-sided limits look like.

We saw two most typical examples when one-sided limits are useful,
expressions *A* into a function and you
run into trouble, you should ask whether the situation could be helped if
*A* is of the one-sided type.

**Example:** What is the outcome of tan(π/2)?

We know that tangent is not defined at

we see that this is exactly the situation where one-sided limits can help. If
we imagine examples of sequences tending to

we should be able to make the correct guess:

^{+}) = −∞,^{-}) = ∞.

So when faced with