Talast (Tusnakt)

Deg Tasleṭ tusnakt, tazmilt n talast teglem tamawaẓt n wazalen n usartu ma d amatar yettruḥu-d s ifeḍ, neɣ d tesɣunt ma d umutti  yettaẓ-d ar waggaz ( yezmer lḥal d yifeḍ ) deg yiri n tɣult n tbadut. ma talast-nni tella deg  tegrumma n waggwaḍ, ihi ad nini d akken asartu neɣ tasɣunt yettaweḍ ( deg waggaz yettwaɣren ) . ma yella mačči akka, ihi d tamhelwayt am wakken yella deg isurta neɣ tisɣunin tiwulanin mačči timezgatin ( am tasɣunt sinus deg +\infin )

ddaw tawtilt n tilin, Akti n talast ad yili s usiḍen gar iferdisen s timhalin tiferdisin, ama nesɛa aṭas n telɣiwin tuffirin i zemren ad ilin. Asnemhel n  tigmi yettaǧǧa mennaw n tikwal ad tekkes tuffrawin-igi.

Aguccel d talast yezmer ad yifsus s tenfalit n ugdazal (ladɣa deg waddad n talast tilemt neɣ tifeḍt ), deg tanasaẓt tamezlagut neɣ deg ifurken  iknuziraren, neɣ taneflit yeffuken neɣ d anasaẓ.

## Talast n usartu

### Imedyaten imezwura s isurta ilawen
ma n d amḍan ameqqran aṭas (ummid), imitti-ines \frac{1}{n} yuẓan aṭas ar 0. wagi nezmer ad negg s tugda yagi : \lim_{n\to\infty} \frac 1n = 0.

Am wakken, nezmer ad naru \lim_{n\to\infty} \sqrt n = +\infty, imi aẓar amkuẓ yezmer ad yili meqqer mliḥ ma ara d yili n meqqar akken iwata (amedya, akken ad nesɛu  \sqrt n > 1000, ilaq ad yili  >  ) 

seg tama nniḍen, ur telli ara talast n usartu awalan ((-1)^n)_{n\in\N}, iwumi ara ad yesɛu s temlellayt azalen 1 d -1, ama ur yettruḥu ara la ar -1 la ar 1, la ar wazal deg uzilal ]-1,1[ neɣ berra-s. imi yella d asartu ugmir, ihi ur yezmir ara ad yruḥ ar +\infty neɣ -\infty

### Agtalɣa

#### talast ifuken
tabadut sɣur Weierstrass n uggwaḍ n usartu ilaw (x_n)_{n\in\N} ar talast ifuken LL\in\R ad nfali s tenfalit agi : 

: \forall \varepsilon>0\quad\exists n_0 \in \N\quad\forall n \geq n_0\quad L-\varepsilon 

neɣ ugar, s tenfalit tagdazalt deg waddad n usartu ilaw, iwumi ara ad yezgu mliḥ deg isurta n wazalen isalalen.

: \forall \varepsilon>0\quad\exists n_0 \in \N\quad\forall n \geq n_0\quad L-\varepsilon 

tabadut agi ad ssiɣzef daɣen ar isurta deg tallunt tamawayt talugant s ubeddel n wazal amagdaz s walugen, arnu s umata ar isurta deg tallunt timizrirt s tenfalit agi : 

: \forall \varepsilon>0\quad\exists n_0 \in \N \quad\forall n \geq n_0\quad \mathrm d(x_n, L) 

deg sin addaden-igi, ur yezmir ara ad illin sin n tlasin i yemgaraden ar yiwen usartu, wagi aɣ-yeǧǧ ad negg tagda yagi :  \lim_{n\to\infty}x_n = L, nettaf tikwal tira yagi x_n \underset{n\to\infty}{\longrightarrow} L 

#### talast timifeḍt
neqqar d akken usartu (x_n)_{n\in\N} yettemhelway ar +\infty deg waddad agi: 

\lim_{n\to\infty}x_n = +\infty \iff \forall M\in\R\quad\exists n_0 \in \N \quad\forall n \geq n_0\quad x_n > M

S terwest, yettemhelway ar -\infty deg waddad agi : 

\lim_{n\to\infty}x_n = -\infty \iff \forall M\in\R\quad\exists n_0 \in \N\quad\forall n \geq n_0\quad x_n 

## Talast n tesɣunt
tazmilt n talast n tesɣunt tettemcabi ar tazmilt n talast n usartu, siwa aya amutti n tesɣunt yezmer ad iruḥ ar wazal menwala deg taɣult n tbadut neɣ deg tilisa-ines.

daɣen, i yal tesɣunt tettwabadu deg yiwen uzilal ]a,b[ \subset \R, nezmer ad nẓer talasin i zemrent ad ilint n tesɣunt deg yal ilaw  c
 deg uzilal, ama daɣen deg yigmiren a
 d b
, igmiren-a zemren ad ilin ifuken neɣ xaṭi. 

### Talast ifuken seg wazal ifuken

#### Tabadut
i yal tasɣunt tilawt neɣ tasemlalt n yiwen umutti ilaw neɣ asemlal, tanfalit n talast ifuken deg wazal ifuken tettemcaba

ar tinna n talast n usartu  

: \lim_{x\to a} f(x) = L \iff\quad\forall \varepsilon \in \R_+^*\quad\exists \delta \in \R_+^*\quad\forall x \in \mathcal D_f\quad(|{x - a}| 

Tella daɣen : 

\lim_ f(x) = L \iff\quad\forall \varepsilon \in \R_+^*\quad\exists \delta \in \R_+^*\quad\forall x \in \mathcal D_f\setminus{\{a\}}\quad(|{x - a}| 

### talast deg uyeffus neɣ d uzelmaḍ

talast n tesɣunt f ar uẓelmaḍ deg ilaw a yettwaru \lim_{x\to a\atop x neɣ \lim_{x\to a^-} f(x)

akken , f(a^-) 

deg a yettwaru \lim_{x\to a\atop x>a} f(x) neɣ\lim_{x\to a+} f(x)

akken, f(a^+)

:

### Imedyaten

* Talast n x-> 1/x deg ifeḍ yegda-d ar 0 : \lim_{x\to \infty} \frac{1}{x} = 0
* talast n x-> 1/x deg 0 ur telli ar. Talast teyeffust d ifeḍ + \infty : \lim_{x\to 0+} \frac{1}{x} = +\infty  \lim_{x\to 0-} \frac{1}{x} = -\infty