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Lektion 48 — Grenzwerte mit trigonometrischen Funktionen

Manipulação de limites trig. Identidades aplicadas a sin x/x, 1−cos x, tan x e variantes.

Used in: 2.º ano do EM (16 anos) · Equiv. Math II japonês cap. 4 · Equiv. Klasse 11 alemã (Grenzwerte trigonometrischer Funktionen)

\lim_{x \to 0} \frac{\sin(kx)}{x} = k, \quad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}

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Rigorous notation, full derivation, hypotheses

Trigonometrische Manipulation

Fundamentale trigonometrische Grenzwerte

Grenzwert	Wert
$\lim_{x \to 0} \sin x / x$	$1$
$\lim_{x \to 0} \sin(kx)/x$	$k$
$\lim_{x \to 0} \tan x / x$	$1$
$\lim_{x \to 0} (1 - \cos x)/x$	$0$
$\lim_{x \to 0} (1 - \cos x)/x^2$	$1/2$
$\lim_{x \to 0} \sin^2 x / x^2$	$1$
$\lim_{x \to 0} \arcsin x / x$	$1$
$\lim_{x \to 0} \arctan x / x$	$1$

Nützliche Identitäten

$1 - \cos x = 2\sin^2(x/2)$
$\sin(a + b) = \sin a \cos b + \cos a \sin b$
$\cos(a + b) = \cos a \cos b - \sin a \sin b$
$1 - \cos x = (1 - \cos^2 x)/(1 + \cos x) = \sin^2 x / (1 + \cos x)$

Techniken

Variablensubstitution $u$ : $\sin(2x)/x = 2 \cdot \sin(2x)/(2x) \to 2$ .
Identitäten zum Umschreiben in Grundform.
Konjugierte: $1 - \cos x = \sin^2 x / (1 + \cos x)$ .
Einschnürung: $-1 \leq \sin x \leq 1$ , multipliziere mit etwas $\to 0$ .

Beweis von $\lim (1 - \cos x)/x^2 = 1/2$

$\frac{1 - \cos x}{x^2} = \frac{2 \sin^2(x/2)}{x^2} = \frac{1}{2} \cdot \frac{\sin^2(x/2)}{(x/2)^2} \to \frac{1}{2} \cdot 1 = \frac{1}{2}$ .

Beweis von $\lim \tan x / x = 1$

$\tan x / x = (\sin x / x) \cdot (1 / \cos x) \to 1 \cdot 1 = 1$ .

Exercise list

40 exercises · 10 with worked solution (25%)

Application 30Understanding 2Modeling 6Challenge 2

Ex. 48.1Application
$\lim_{x \to 0} \sin(7x)/x$ . (Antw.: 7.)
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Ex. 48.2Application
$\lim_{x \to 0} \sin(3x)/\sin(5x)$ . (Antw.: $3/5$ .)
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Ex. 48.3Application
$\lim_{x \to 0} \tan(2x)/\sin(3x)$ . (Antw.: $2/3$ .)
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Ex. 48.4Application
$\lim_{x \to 0} \sin^2 x / x$ . (Antw.: 0.)
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Ex. 48.5ApplicationAnswer key
$\lim_{x \to 0} (1 - \cos(2x))/x^2$ . (Antw.: 2.)
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Ex. 48.6Application
$\lim_{x \to \pi/2} (1 - \sin x)/(\pi/2 - x)^2$ . (Antw.: $1/2$ .)
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Ex. 48.7Application
$\lim_{x \to 0} (\tan x - \sin x)/x^3$ . (Antw.: $1/2$ .)
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Ex. 48.8ApplicationAnswer key
$\lim_{x \to 0} \arcsin(2x)/x$ . (Antw.: 2.)
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Ex. 48.9Application
$\lim_{x \to 0} \arctan(3x)/\sin(2x)$ . (Antw.: $3/2$ .)
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Ex. 48.10Application
$\lim_{x \to \pi} \sin x/(x - \pi)$ . (Antw.: $-1$ .)
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Ex. 48.11Application
$\lim_{x \to 0} (\cos x - 1)/(\sin x)$ . (Antw.: 0.)
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Ex. 48.12Application
$\lim_{x \to 0} x \cdot \cot x$ . (Antw.: 1.)
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Ex. 48.13Application
$\lim_{x \to 0} \sin(x^3)/x$ . (Antw.: 0.)
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Ex. 48.14ApplicationAnswer key
$\lim_{x \to 0} \sin(x)\sin(2x)/x^2$ . (Antw.: 2.)
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Ex. 48.15ApplicationAnswer key
$\lim_{x \to 0} (1 - \cos(3x))/(1 - \cos(2x))$ . (Antw.: $9/4$ .)
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Ex. 48.16Application
$\lim_{x \to 0} (1 - \cos x)/x$ . (Antw.: 0.)
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Ex. 48.17ApplicationAnswer key
$\lim_{x \to 0} \sin(5x) \cdot \cot(3x)$ . (Antw.: $5/3$ .)
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Ex. 48.18ApplicationAnswer key
$\lim_{x \to 0} (\cos x - \cos(3x))/x^2$ . (Antw.: 4.)
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Ex. 48.19Application
$\lim_{x \to 0} \sin(\pi x)/\sin(\pi(1-x))$ .
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Ex. 48.20Application
$\lim_{x \to 0} \sin(\sin x)/x$ . (Antw.: 1.)
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Ex. 48.21Application
$\lim_{x \to \pi/4} (\tan x - 1)/(x - \pi/4)$ . (Antw.: 2.)
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Ex. 48.22ApplicationAnswer key
$\lim_{x \to \pi/2} \cos x / (x - \pi/2)$ . (Antw.: $-1$ .)
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Ex. 48.23Application
$\lim_{x \to 0} (\sec x - 1)/x^2$ . (Antw.: $1/2$ .)
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Ex. 48.24Application
$\lim_{x \to 0} (\tan^2 x)/x^2$ . (Antw.: 1.)
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Ex. 48.25Application
$\lim_{x \to 0} (1 - \cos x \cdot \cos(2x))/x^2$ . (Antw.: $5/2$ .)
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Ex. 48.26ApplicationAnswer key
$\lim_{x \to 0} \sin(\pi - x)/x$ .
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Ex. 48.27Application
$\lim_{x \to 0} (\sin(a+x) - \sin a)/x$ . (Antw.: $\cos a$ .)
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Ex. 48.28Application
$\lim_{x \to 0} (\cos(a+x) - \cos a)/x$ . (Antw.: $-\sin a$ .)
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Ex. 48.29Application
$\lim_{x \to 0} \sin x \cdot \ln(1+x)/x^2$ . (Antw.: 1.)
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Ex. 48.30Application
$\lim_{x \to 0} (e^x - \cos x)/x$ . (Antw.: 1.)
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Ex. 48.31ModelingAnswer key
In der Regelung: $H(s)$ mit Pol bei $j\omega$ — Grenzwert $\to \infty$ in der Resonanz.
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Ex. 48.32Modeling
Approximation $\sin x \approx x - x^3/6$ für kleine Winkel. Relativer Fehler bei $x = 0{,}1$ ?
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Ex. 48.33Modeling
In der Mechanik: Pendel $\ddot \theta = -(g/L) \sin \theta \approx -(g/L)\theta$ für kleine. Begründe via Grenzwert.
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Ex. 48.34Modeling
Brechung: $n_1 \sin \theta_1 = n_2 \sin \theta_2$ . Paraxiale Approximation $n_1 \theta_1 \approx n_2 \theta_2$ .
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Ex. 48.35Modeling
In AM-Signal: $V(t) = (1 + m \cos(\omega_m t)) \cos(\omega_c t)$ . Für kleines $m$ entwickle via Grenzwert.
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Ex. 48.36Modeling
Beugung an einem Spalt: Muster $\propto (\sin u/u)^2$ mit $u = \pi a \sin\theta/\lambda$ . Berechne Grenzwert in $\theta = 0$ .
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Ex. 48.37Understanding
Zeige $\lim_{x \to 0} \sin(kx)/x = k$ über Variablenwechsel.
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Ex. 48.38Understanding
Zeige $(1 - \cos x)/x^2 \to 1/2$ unter Verwendung der Identität $1 - \cos x = 2\sin^2(x/2)$ .
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Ex. 48.39Challenge
$\lim_{x \to 0} (\sin(\sin x) - x)/x^3$ .
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Ex. 48.40ChallengeAnswer key
$\lim_{x \to 0} (\cos(\sin x) - \cos x)/x^4$ .
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Quellen

Active Calculus — Boelkins · 2024 · §1.8.
APEX Calculus — Hartman · 2024 · §1.5.
Calculus (Volume 1) — OpenStax · 2016 · §2.6.