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 + ∞ {\displaystyle +\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 {\displaystyle n} d amḍan ameqqran aṭas (ummid), imitti-ines 1 n {\displaystyle {\frac {1}{n}}} yuẓan aṭas ar 0. wagi nezmer ad negg s tugda yagi : lim n → ∞ 1 n = 0 {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}=0} .

Am wakken, nezmer ad naru lim n → ∞ n = + ∞ {\displaystyle \lim _{n\to \infty }{\sqrt {n}}=+\infty } , imi aẓar amkuẓ yezmer ad yili meqqer mliḥ ma ara d yili n {\displaystyle n} meqqar akken iwata (amedya, akken ad nesɛu n > 1000 {\displaystyle {\sqrt {n}}>1000} , ilaq ad yili n > 1 000 000 )

seg tama nniḍen, ur telli ara talast n usartu awalan ( ( − 1 ) n ) n ∈ N {\displaystyle ((-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 + ∞ {\displaystyle +\infty } neɣ − ∞ {\displaystyle -\infty }

Agtalɣa

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

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

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 :

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 → ∞ x n = L {\displaystyle \lim _{n\to \infty }x_{n}=L} , nettaf tikwal tira yagi x n ⟶ n → ∞ L {\displaystyle x_{n}{\underset {n\to \infty }{\longrightarrow }}L}

neqqar d akken usartu ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} yettemhelway ar + ∞ {\displaystyle +\infty } deg waddad agi:

lim n → ∞ x n = + ∞ ⟺ ∀ M ∈ R ∃ n 0 ∈ N ∀ n ≥ n 0 x n > M {\displaystyle \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 − ∞ {\displaystyle -\infty } deg waddad agi :

lim n → ∞ x n = − ∞ ⟺ ∀ M ∈ R ∃ n 0 ∈ N ∀ n ≥ n 0 x n < M {\displaystyle \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 [ ⊂ R {\displaystyle ]a,b[\subset \mathbb {R} } , nezmer ad nẓer talasin i zemrent ad ilint n tesɣunt deg yal ilaw c {\displaystyle c} deg uzilal, ama daɣen deg yigmiren a {\displaystyle a} d b {\displaystyle b} , igmiren-a zemren ad ilin ifuken neɣ xaṭi.

Talast ifuken seg wazal ifuken

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

Tella daɣen :

lim x → a x ≠ a f ( x ) = L ⟺ ∀ ε ∈ R + ∗ ∃ δ ∈ R + ∗ ∀ x ∈ D f ∖ { a } ( | x − a | < δ ⇒ | f ( x ) − L | < ε ) {\displaystyle \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

talast n tesɣunt f ar uẓelmaḍ deg ilaw a yettwaru lim x → a x < a f ( x ) {\displaystyle \lim _{x\to a \atop x<a}f(x)} neɣ lim x → a − f ( x ) {\displaystyle \lim _{x\to a^{-}}f(x)}

akken , f ( a − ) {\displaystyle f(a^{-})}

deg a yettwaru lim x → a x > a f ( x ) {\displaystyle \lim _{x\to a \atop x>a}f(x)} neɣ lim x → a + f ( x ) {\displaystyle \lim _{x\to a+}f(x)}

akken, f ( a + ) {\displaystyle f(a^{+})}

Imedyaten