Analysis I - IV by Hilgert J.

By Hilgert J.

Der vorliegende textual content ist eine vorlaufige Ausarbeitung meiner Vorlesungen research I-IV (Wintersemester 2004/2005 { Sommersemester 2006) an der Universitat Paderborn.

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B. f 1 n+r ✲ ¨ = r f¨ ur alle r ∈ [0, 1[ (vgl. 15). 3 : Betrachte die Funktion f : R → R, x → 1 x2 +1 . 1. GRENZWERTE VON FUNKTIONEN 35 ✻ ........................................ .......... ............ ........................ . ✲ Wir zeigen, daß limx→1 f (x) = 12 . Dazu sei > 0. Wir m¨ ussen eine Zahl δ > 0 finden, f¨ ur die gilt ∀ 0 < |x − 1| < δ : 1 1 − < . x2 + 1 2 Es gilt 1 1+x 1 − = |x − 1|. A. δ < 1 - ohne Beschr¨ankung der Allgemeinheit).

1 −1 (ii) limh→0 x+hh x = − x12 . Hier ist der Bruch bei festem x als Funktion in h zu betrachten, die wiederum sowohl in einem offenen Intervall mit rechtem Rand 0 als auch in einem Intervall mit linkem Rand 0 definiert ist. 6 : Seien f und g reellwertige Funktionen, die auf einer Teilmenge M von R definiert sind und x0 ∈ R erf¨ ulle eine der drei folgenden Bedingungen   ∀δ > 0 : ]x0 , x0 + δ[ ∩M = ∅, ∀δ > 0 : ]x0 − δ, x0 [ ∩M = ∅,   ∀δ > 0 : ]x0 − δ, x0 + δ[ ∩M = ∅.

Grenzwert ✲ a b .. .. .. ... ... ... ...... ......... ............... 1. 1 : (i) Die Funktion f : ]2, ∞[→ R, −(x − 2)2 |x − 2| + (x − 2) x→ hat in 2 den rechtsseitigen Grenzwert 0. y ✻ ❍ 2 ❍❍ ✲x ❍❍ (ii) F¨ ur a ∈ R bezeichnet man mit a die gr¨oßte ganze Zahl, die kleiner oder gleich a ist. Man nennt a den ganzzahligen Anteil von a. Dann hat die Funktion f : ]0, ∞[→ R, x→x 2 x + −2 x 2 in 0 den rechtsseitigen Grenzwert 0.

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