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 $+\infty$ )

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 : $\lim _{n\to \infty }{\frac {1}{n}}=0$.

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 n > 1 000 000 )

seg tama nniḍen, ur telli ara talast n usartu awalan $((-1)^{n})_{n\in \mathbb {N} }$, 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 $(x_{n})_{n\in \mathbb {N} }$ ar talast ifuken L$L\in \mathbb {R}$ ad nfali s tenfalit agi :

$\forall \varepsilon >0\quad \exists n_{0}\in \mathbb {N} \quad \forall n\geq n_{0}\quad L-\varepsilon <x_{n}<L+\varepsilon$

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

$\forall \varepsilon >0\quad \exists n_{0}\in \mathbb {N} \quad \forall n\geq n_{0}\quad L-\varepsilon <x_{n}<L+\varepsilon$

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 :

$\forall \varepsilon >0\quad \exists n_{0}\in \mathbb {N} \quad \forall n\geq n_{0}\quad \mathrm {d} (x_{n},L)<\varepsilon$

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 $(x_{n})_{n\in \mathbb {N} }$ yettemhelway ar $+\infty$ deg waddad agi:

$\lim _{n\to \infty }x_{n}=+\infty \iff \forall M\in \mathbb {R} \quad \exists n_{0}\in \mathbb {N} \quad \forall n\geq n_{0}\quad x_{n}>M$

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

$\lim _{n\to \infty }x_{n}=-\infty \iff \forall M\in \mathbb {R} \quad \exists n_{0}\in \mathbb {N} \quad \forall n\geq n_{0}\quad x_{n}<M$

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 $]a,b[\subset \mathbb {R}$, 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

$\lim _{x\to a}f(x)=L\iff \quad \forall \varepsilon \in \mathbb {R} _{+}^{*}\quad \exists \delta \in \mathbb {R} _{+}^{*}\quad \forall x\in {\mathcal {D}}_{f}\quad (|{x-a}|<\delta \Rightarrow |f(x)-L|<\varepsilon )$

Tella daɣen :

$\lim _{{x\to a} \atop {x\neq a}}f(x)=L\iff \quad \forall \varepsilon \in \mathbb {R} _{+}^{*}\quad \exists \delta \in \mathbb {R} _{+}^{*}\quad \forall x\in {\mathcal {D}}_{f}\setminus {\{a\}}\quad (|{x-a}|<\delta \Rightarrow |f(x)-L|<\varepsilon )$

talast deg uyeffus neɣ d uzelmaḍ

talasin deg yigrimiren n uzilal n tesɣunt timittit: $\lim _{x\to -\infty }{\tfrac {1}{x}}=\lim _{x\to +\infty }{\tfrac {1}{x}}=0$, $\lim _{x\to 0^{-}}{\tfrac {1}{x}}=-\infty$, $\lim _{x\to 0^{+}}{\tfrac {1}{x}}=+\infty$

talast n tesɣunt f ar uẓelmaḍ deg ilaw a yettwaru $\lim _{x\to a \atop x<a}f(x)$ 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 $+\infty$ : $\lim _{x\to 0+}{\frac {1}{x}}=+\infty$ $\lim _{x\to 0-}{\frac {1}{x}}=-\infty$