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The limit of sequences and functions is very important in mathematics. In addition, it requires regular practice of various exercises on the chapter.

## I. Suites and functions: comparative study.

Remark:

The numerical sequences being particular functions (defined on N or a part of N ), one will find necessarily for the sequences and the functions, analogous properties.

Not all properties will be listed here, but only those whose comparison is instructive.

The sequences will appear in the right column and the numerical functions in the left column. Except in special cases, sequences are defined on N or possibly a part of N (from a certain rank ) and functions are defined on R or a part of R (most of the time on an interval I ).

** 1 Sense of variation of functions and numerical sequences. **

f increasing on I For all aI and bI, if a b then f(a) f(b)
f strictly increasing on I For all aI and bI, if a < b then f(a) < f(b) f decreasing on I For all aI and bI, if a b then f(a) f(b) f strictly decreasing on I For all aI and bI, if a < b then f(a) > f(b) f constant on I For all aI and bI, if a b then f(a) = f(b) f monotonic on I f keeps the same direction of variation on the interval I If f is derivable on I, then: f increasing on I For any xI, f ‘ (x) 0 If f is differentiable on I, then: f decreasing on I For any xI, f ‘ (x) 0 If f is derivable on I, then: f constant on I For any xI, f ‘ (x) = 0 |
(_{Un}) increasing For all nN , we have:_{Un} _{Un+1}
( ( ( ( ( |

Note:

It may be that this is not true for the whole set N : specify then from which rank this is true.

### 2. major, minor and bounded sequences and functions.

f major on I There exists MR, such that, for any xI, we have: f(x) M
f minor on I There exists mR, such that, for any xI, we have: f(x) m f bounded on I f is increased and decreased on I |
(_{Un}) increased There exists MR, such that, for any nN, we have:_{Un} M
( ( |

## II. limits of sequences and functions.

**1.finite limit (l ****) and the****imite in + of functions and sequences.**

**2.finite limit in ****– **** of numerical functions.**

If f is defined on R or an interval of the form ]- ; A] where AR and if any open interval containing also contains all the values of f(x) for – x large enough, we say that f tends to when x tends to – .

We write

In this situation, the graphical representation of f has a **horizontal asymptote** of equation y = in – .

Example:

.

### 3. finite limit of a function at a point aR

If f is defined on R or an interval I containing aR and if any open interval containing also contains all values of f(x) for any xI close enough to a, we say that f tends to when x tends to a.

We write .

Example:

**II.case where the function or the sequence has no limit.**

Without going into theoretical details, we will give some examples of functions and sequences that do not have a limit. It is interesting to visualize these examples on a graphing calculator.

1. f defined on by .

We have: If x > 0, f(x) = 1, so and if x < 0, f(x) = -1, so .

The limits to the left and to the right of 0 exist, but are different so** f has no limit in 0**.

2. For any nN,_{Un} = sin n takes an infinity of values on the interval [-1 ; 1], without ever approaching a limit value.

3. For any nN, takes alternatively the values -1 and 1, thus no limit.

**III. The comparison theorems **

For functions, in the properties below, the letter a designates a real number as well as +. or – .

When a = + The functions are defined on R or an interval I of the form [ A ; + ]. [ where A is a real number.

When a = – The functions are defined on R or an interval I of the form ] – ; A ] where A is a real number.

When a R , functions are defined on R or an interval I of the form [ A ; B ] where A and B are real and a[ A ; B ].

If the limit in question is the left-hand limit of a, the functions are defined on an interval I of the form ] – ; a [ or [ A ; a [ where A is a real number.

If the limit concerned is the limit to the right of a, the functions are defined on an interval I of the form ] a ; + [ or ] a ; A ] where A is a real number.

For sequences, the index n is a natural number greater than or equal to a certain rank (which will often be 0).

**IV. Composition theorem of two functions****:**

If f is a function defined on an interval J such that, for any xI, we have:

y = u(x)J, i.e.: u(I) J.

If we also have and , then .

If f is a function defined on an interval J such that:

for any integer we have:

If we also have and , then .

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.If f is defined on R or an interval of the form [A; +], then[ where AR and if any open interval containing also contains all values of f(x) for x large enough, we say that f tends to when x tends to +. .we write .
In this situation, the graphical representation of f has a Example: | If any open interval containing also contains all the terms of the sequence (Un) from a certain rank, we say that (_{Un}) converges to . We write
Example: |