List of issues considered in the topic:
1) determination of the sequence limit;
2) basic properties of limits;
3) types of uncertainties and ways to eliminate them;
4) rules for calculating the limits of the infinity function.
Glossary on the topic
A numbered series of numbers a1, a2,..., ap,... is called a numerical sequence.
A numerical sequence is called an ascending sequence if each of its members is larger than the previous one, in other words, if there is a true inequality for any of them (similarly, a descending numerical sequence is defined).
A sequence is called monotonous if it is either increasing or decreasing.
A sequence a1, a2, ..., ... is called limited if a common boundary can be specified for its members, i.e. if there is such a number C that the inequality is performed for all numbers n.
The sequence is bounded at the bottom, if there is a number m such that inequality is executed for any n . The number m is called the lower boundary of the sequence.
The number b is called the limit of the sequence if any pre-selected neighborhood of point b contains all the members of the sequence starting with some number.
When calculating the limits, the expressions with undefined values often appear. Such expressions are called uncertainties.
Basic Literature:
Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E., etc., edited by Zhizhchenko A.B. Algebra and the beginning of mathematical analysis (basic and profile levels) 11 cells - Moscow: Enlightenment, 2014.
Additional literature:
Shabunin M.I., Tkacheva M.V., Fedorova N.E. Didactic materials of Algebra and the beginning of mathematical analysis (basic and profile levels) 11 ph. - M.: Enlightenment, 2017.
Theoretical material for independent study
Let each natural number be matched with a real number: number 1 corresponds to number a, number 2 corresponds to number a2, ..., number n corresponds to number a, etc. Then they say that a numeric sequence is given, and they write a1, a2, ..., ap or (ap), where a1, a2, ..., ..., a - members of the sequence.
A measured series of numbers a1, a2, ..., ap, ... is called a numeric sequence.
Methods of assigning sequences.
The simplest way to set a sequence is to set it using the formula of the common term, i.e., the formula that explicitly expresses the dependence of the n-th term of the sequence on n.
For example, the formula ап=2n specifies a sequence of even numbers 2,4,6,8,... .
Another important way to specify a sequence is the recurrence method, in which an expression linking the n-th term of the sequence to one or more previous ones is specified.
The word recurrence comes from the Latin word recurrens, which means "return". When we calculate a new, new sequence member, we sort of go back and use the already calculated previous members.
For example, the recurrence ratio an=an-1+2 together with equation a1=1 defines an arithmetic progression with the first term 1 and a difference of 2:1, 3, 5, 7, ... . This is nothing more than a sequence of odd numbers.
The sequence can also be given by a verbal description, which defines the process of building the sequence members.
So try this limit calculator