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 $+\mathrm{\infty }$ )

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

Am wakken, nezmer ad naru $\underset{n\to \mathrm{\infty }}{\lim }\sqrt{n}=+\mathrm{\infty }$, 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 Talɣa:Mvar > 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 $+\mathrm{\infty }$ neɣ $-\mathrm{\infty }$

Agtalɣa

talast ifuken

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

$\mathrm{\forall }\epsilon >0\mathrm{\exists }{n}_0\in \mathbb{N}\mathrm{\forall }n\ge n_{0}L-\epsilon <x_n<L+\epsilon$

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

$\mathrm{\forall }\epsilon >0\mathrm{\exists }{n}_0\in \mathbb{N}\mathrm{\forall }n\ge n_{0}L-\epsilon <x_n<L+\epsilon$

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 :

$\mathrm{\forall }\epsilon >0\mathrm{\exists }{n}_0\in \mathbb{N}\mathrm{\forall }n\ge n_0\mathrm{d}(x_n,L)<\epsilon$

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

talast timifeḍt

neqqar d akken usartu $(x_n{)}_{n\in \mathbb{N}}$ yettemhelway ar $+\mathrm{\infty }$ deg waddad agi:

$\underset{n\to \mathrm{\infty }}{\lim }{x}_n=+\mathrm{\infty }⟺\mathrm{\forall }M\in ℝ\mathrm{\exists }{n}_0\in \mathbb{N}\mathrm{\forall }n\ge n_0{x}_n>M$

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

$\underset{n\to \mathrm{\infty }}{\lim }{x}_n=-\mathrm{\infty }⟺\mathrm{\forall }M\in ℝ\mathrm{\exists }{n}_0\in \mathbb{N}\mathrm{\forall }n\ge n_0{x}_n<M$

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 $]a,b[\subset ℝ$, 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

$\underset{x\to a}{\lim }f(x)=L⟺\mathrm{\forall }\epsilon \in {ℝ}_{+}^{*}\mathrm{\exists }\delta \in {ℝ}_{+}^{*}\mathrm{\forall }x\in {𝒟}_f(|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon )$

Tella daɣen :

$\underset{\genfrac{}{}{0ex}{}{x\to a}{x\ne a}}{\lim }f(x)=L⟺\mathrm{\forall }\epsilon \in {ℝ}_{+}^{*}\mathrm{\exists }\delta \in {ℝ}_{+}^{*}\mathrm{\forall }x\in {𝒟}_f\setminus \{a\}(|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon )$

talast deg uyeffus neɣ d uzelmaḍ

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

talast n tesɣunt f ar uẓelmaḍ deg ilaw a yettwaru $\underset{\genfrac{}{}{0ex}{}{x\to a}{x<a}}{\lim }f(x)$ neɣ $\underset{x\to a^{-}}{\lim }f(x)$

akken , $f(a^{-})$

deg a yettwaru $\underset{\genfrac{}{}{0ex}{}{x\to a}{x>a}}{\lim }f(x)$ neɣ$\underset{x\to a+}{\lim }f(x)$

akken, $f(a^{+})$

Imedyaten

• Talast n x-> 1/x deg ifeḍ yegda-d ar 0 : $\underset{x\to \mathrm{\infty }}{\lim }\frac{1}{x}=0$
• talast n x-> 1/x deg 0 ur telli ar. Talast teyeffust d ifeḍ $+\mathrm{\infty }$ : $\underset{x\to 0+}{\lim }\frac{1}{x}=+\mathrm{\infty }$ $\underset{x\to 0-}{\lim }\frac{1}{x}=-\mathrm{\infty }$