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Lektion 45 — Bemerkenswerte Grenzwerte und unbestimmte Formen

lim sin x/x = 1, lim (1+1/n)^n = e, lim (e^x − 1)/x = 1. Indeterminações e técnicas.

Used in: 2.º ano EM (Trim. 5) · Equiv. Math II japonês (cap. 3 — limites especiais) · Equiv. Klasse 11 alemã (Grenzwerte trigonometrisch) · Equiv. H2 Math singapurense (Special limits)

lim⁡x→0sin⁡xx=1,lim⁡x→∞(1+1x)x=e,lim⁡x→0ex−1x=1\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e, \quad \lim_{x \to 0} \frac{e^x - 1}{x} = 1
Choose your door

Rigorous notation, full derivation, hypotheses

Bemerkenswerte Grenzwerte und Techniken

Die 3 fundamentalen

  1. lim⁡x→0sin⁡xx=1\displaystyle\lim_{x \to 0} \frac{\sin x}{x} = 1 (geometrisch — Einschnürung).
  2. lim⁡x→∞(1+1x)x=e\displaystyle\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e (definiert ee).
  3. lim⁡x→0ex−1x=1\displaystyle\lim_{x \to 0} \frac{e^x - 1}{x} = 1 (folgt aus dem vorherigen).

Nützliche Varianten (Tabelle)

GrenzwertWert
lim⁡x→0sin⁡x/x\lim_{x \to 0} \sin x / x11
lim⁡x→0sin⁡(kx)/x\lim_{x \to 0} \sin(kx)/xkk
lim⁡x→0tan⁡x/x\lim_{x \to 0} \tan x / x11
lim⁡x→0(1−cos⁡x)/x2\lim_{x \to 0} (1-\cos x)/x^21/21/2
lim⁡x→0(1−cos⁡x)/x\lim_{x \to 0} (1-\cos x)/x00
lim⁡x→0arcsin⁡x/x\lim_{x \to 0} \arcsin x / x11
lim⁡x→0arctan⁡x/x\lim_{x \to 0} \arctan x / x11
lim⁡x→0(ex−1)/x\lim_{x \to 0} (e^x - 1)/x11
lim⁡x→0(ax−1)/x\lim_{x \to 0} (a^x - 1)/xln⁡a\ln a
lim⁡x→0ln⁡(1+x)/x\lim_{x \to 0} \ln(1+x)/x11
lim⁡x→0((1+x)a−1)/x\lim_{x \to 0} ((1+x)^a - 1)/xaa
lim⁡x→∞(1+a/x)x\lim_{x \to \infty} (1 + a/x)^xeae^a
lim⁡x→0+xx\lim_{x \to 0^+} x^x11
lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x00

Häufige unbestimmte Formen (7)

0/00/0, ∞/∞\infty/\infty, ∞−∞\infty - \infty, 0⋅∞0 \cdot \infty, 1∞1^\infty, 000^0, ∞0\infty^0 — alle durch Manipulation oder L'Hôpital lösbar (Lektion 64).

Techniken

  • Faktorisieren und kürzen (0/00/0 polynomial).
  • Konjugiertes (0/00/0 mit Wurzeln).
  • Trigonometrisch: sin⁡x≈x\sin x \approx x (xx klein).
  • Variable wechseln: u=1/xu = 1/x, u=x−au = x - a.
  • Logarithmus: AB=eBln⁡AA^B = e^{B \ln A} wandelt 1∞1^\infty, 000^0 in 0/00/0 um.

Exercise list

42 exercises · 10 with worked solution (25%)

Application 30Understanding 4Modeling 6Challenge 2
  1. Ex. 45.1Application
    lim⁡x→0sin⁡(2x)/x\lim_{x \to 0} \sin(2x)/x. (Antw.: 2.)
    Solve online
  2. Ex. 45.2Application
    lim⁡x→0sin⁡(5x)/sin⁡(3x)\lim_{x \to 0} \sin(5x)/\sin(3x). (Antw.: 5/35/3.)
    Solve online
  3. Ex. 45.3Application
    lim⁡x→0tan⁡x/x\lim_{x \to 0} \tan x / x.
    Solve online
  4. Ex. 45.4Application
    lim⁡x→0(1−cos⁡x)/x2\lim_{x \to 0} (1 - \cos x)/x^2. (Antw.: 1/21/2.)
    Solve online
  5. Ex. 45.5Application
    lim⁡x→∞(1+2/x)x\lim_{x \to \infty} (1 + 2/x)^x. (Antw.: e2e^2.)
    Solve online
  6. Ex. 45.6Application
    lim⁡x→0(1+3x)1/x\lim_{x \to 0} (1 + 3x)^{1/x}.
    Solve online
  7. Ex. 45.7Application
    lim⁡x→0(e2x−1)/x\lim_{x \to 0} (e^{2x} - 1)/x. (Antw.: 2.)
    Solve online
  8. Ex. 45.8ApplicationAnswer key
    lim⁡x→0(ex−1)/sin⁡x\lim_{x \to 0} (e^x - 1)/\sin x.
    Solve online
  9. Ex. 45.9Application
    lim⁡x→0ln⁡(1+x)/x\lim_{x \to 0} \ln(1+x)/x.
    Solve online
  10. Ex. 45.10ApplicationAnswer key
    lim⁡x→0ln⁡(1+5x)/x\lim_{x \to 0} \ln(1 + 5x)/x.
    Solve online
  11. Ex. 45.11Application
    lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x. (Antw.: 0.)
    Solve online
  12. Ex. 45.12ApplicationAnswer key
    lim⁡x→0+xx\lim_{x \to 0^+} x^x. (Antw.: 1.)
    Solve online
  13. Ex. 45.13Application
    lim⁡x→1(1/(1−x)−2/(1−x2))\lim_{x \to 1} (1/(1-x) - 2/(1-x^2)). (Antw.: −1/2-1/2.)
    Solve online
  14. Ex. 45.14Application
    lim⁡x→0sin⁡(x2)/x\lim_{x \to 0} \sin(x^2)/x.
    Solve online
  15. Ex. 45.15ApplicationAnswer key
    lim⁡x→∞(1−1/x)x\lim_{x \to \infty} (1 - 1/x)^x. (Antw.: 1/e1/e.)
    Solve online
  16. Ex. 45.16ApplicationAnswer key
    lim⁡x→0((1+x)5−1)/x\lim_{x \to 0} ((1+x)^5 - 1)/x. (Antw.: 5.)
    Solve online
  17. Ex. 45.17Application
    lim⁡x→0(3x−1)/x\lim_{x \to 0} (3^x - 1)/x. (Antw.: ln⁡3\ln 3.)
    Solve online
  18. Ex. 45.18Application
    lim⁡x→0(arcsin⁡x)/x\lim_{x \to 0} (\arcsin x)/x.
    Solve online
  19. Ex. 45.19Application
    lim⁡x→0(arctan⁡x)/x\lim_{x \to 0} (\arctan x)/x.
    Solve online
  20. Ex. 45.20Application
    lim⁡x→∞(x/(x+1))x\lim_{x \to \infty} (x/(x+1))^x. (Antw.: 1/e1/e.)
    Solve online
  21. Ex. 45.21Application
    lim⁡x→0(cos⁡x)1/x2\lim_{x \to 0} (\cos x)^{1/x^2}. (Antw.: e−1/2e^{-1/2}.)
    Solve online
  22. Ex. 45.22Application
    lim⁡x→0(cos⁡x)1/x\lim_{x \to 0} (\cos x)^{1/x}. (Antw.: 1.)
    Solve online
  23. Ex. 45.23ApplicationAnswer key
    lim⁡x→0((sin⁡x)/x)1/x2\lim_{x \to 0} ((\sin x)/x)^{1/x^2}.
    Solve online
  24. Ex. 45.24Application
    lim⁡x→∞((x+2)/(x−1))x\lim_{x \to \infty} ((x+2)/(x-1))^x. (Antw.: e3e^3.)
    Solve online
  25. Ex. 45.25ApplicationAnswer key
    lim⁡x→0+xsin⁡x\lim_{x \to 0^+} x^{\sin x}. (Antw.: 1.)
    Solve online
  26. Ex. 45.26Application
    lim⁡x→0+(sin⁡x)x\lim_{x \to 0^+} (\sin x)^x. (Antw.: 1.)
    Solve online
  27. Ex. 45.27Application
    lim⁡x→1x1/(x−1)\lim_{x \to 1} x^{1/(x-1)}.
    Solve online
  28. Ex. 45.28Application
    lim⁡x→0(1+sin⁡x)1/x\lim_{x \to 0} (1 + \sin x)^{1/x}.
    Solve online
  29. Ex. 45.29Application
    lim⁡x→0(ex+x)1/x\lim_{x \to 0} (e^x + x)^{1/x}.
    Solve online
  30. Ex. 45.30Application
    lim⁡n→∞(n+1)/n2⋅sin⁡(n)\lim_{n \to \infty} (n+1)/n^2 \cdot \sin(n). (Antw.: 0 via Einschnürung.)
    Solve online
  31. Ex. 45.31Modeling
    Stetige Verzinsung: V0=1000V_0 = 1000, r=5%r = 5\%/Jahr, T=10T = 10 Jahre. Berechne V(T)V(T) über lim⁡n→∞V0(1+r/n)nT\lim_{n \to \infty} V_0(1 + r/n)^{nT}.
    Solve online
  32. Ex. 45.32Modeling
    Radioaktiver Zerfall: Halbwertszeit 5 Jahre. N(t)/N0N(t)/N_0 nach 12 Jahren? Verwende e−λte^{-\lambda t} mit λ=ln⁡2/5\lambda = \ln 2 / 5.
    Solve online
  33. Ex. 45.33ModelingAnswer key
    Kleines Pendel: T=2πL/gT = 2\pi\sqrt{L/g}. Begründe Substitution sin⁡θ≈θ\sin \theta \approx \theta über lim⁡sin⁡x/x=1\lim \sin x/x = 1.
    Solve online
  34. Ex. 45.34Modeling
    Optische Approximation: für kleine Winkel sin⁡θ≈θ\sin \theta \approx \theta, tan⁡θ≈θ\tan \theta \approx \theta. Relativer Fehler bei θ=5°\theta = 5°?
    Solve online
  35. Ex. 45.35Modeling
    Poisson-Verteilung seltener Ereignisse: lim⁡n→∞,np=λ(nk)pk(1−p)n−k=e−λλk/k!\lim_{n \to \infty, np = \lambda} \binom{n}{k} p^k (1-p)^{n-k} = e^{-\lambda}\lambda^k/k!.
    Solve online
  36. Ex. 45.36ModelingAnswer key
    In der Finanzwelt: kurzlaufende amerikanische Option, Payoff approximiert durch S0⋅σTS_0 \cdot \sigma \sqrt T über Entwicklung sin⁡/(⋅)→1\sin/(\cdot) \to 1.
    Solve online
  37. Ex. 45.37Understanding
    Beweise lim⁡sin⁡x/x=1\lim \sin x / x = 1 über Einheitskreis und Einschnürung.
    Solve online
  38. Ex. 45.38UnderstandingAnswer key
    Zeige, dass lim⁡(1+1/x)x\lim (1 + 1/x)^x für x∈Nx \in \mathbb{N} monoton wachsend ist.
    Solve online
  39. Ex. 45.39Understanding
    Beweise lim⁡(ex−1)/x=1\lim (e^x - 1)/x = 1 über Taylor-Reihe.
    Solve online
  40. Ex. 45.40Understanding
    Zeige lim⁡x→0(ax−1)/x=ln⁡a\lim_{x \to 0} (a^x - 1)/x = \ln a.
    Solve online
  41. Ex. 45.41Challenge
    lim⁡x→0(tan⁡x−sin⁡x)/x3\lim_{x \to 0} (\tan x - \sin x)/x^3. (Antw.: 1/21/2.)
    Solve online
  42. Ex. 45.42Challenge
    lim⁡x→∞x(ln⁡(x+1)−ln⁡x)\lim_{x \to \infty} x(\ln(x+1) - \ln x). (Antw.: 1.)
    Solve online

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Updated on 2026-04-30 · Author(s): Clube da Matemática

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