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Lesson 48 — Limits with Trigonometric Functions

Manipulating trig limits. Identities applied to sin x/x, 1−cos x, tan x, and variants.

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

lim⁡x→0sin⁡(kx)x=k,lim⁡x→01−cos⁡xx2=12\lim_{x \to 0} \frac{\sin(kx)}{x} = k, \quad \lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}
Choose your door

Rigorous notation, full derivation, hypotheses

Trig manipulation

Fundamental trigonometric limits

LimitValue
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)/x\lim_{x \to 0} (1 - \cos x)/x00
lim⁡x→0(1−cos⁡x)/x2\lim_{x \to 0} (1 - \cos x)/x^21/21/2
lim⁡x→0sin⁡2x/x2\lim_{x \to 0} \sin^2 x / x^211
lim⁡x→0arcsin⁡x/x\lim_{x \to 0} \arcsin x / x11
lim⁡x→0arctan⁡x/x\lim_{x \to 0} \arctan x / x11

Useful identities

  • 1−cos⁡x=2sin⁡2(x/2)1 - \cos x = 2\sin^2(x/2)
  • sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin(a + b) = \sin a \cos b + \cos a \sin b
  • cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a + b) = \cos a \cos b - \sin a \sin b
  • 1−cos⁡x=(1−cos⁡2x)/(1+cos⁡x)=sin⁡2x/(1+cos⁡x)1 - \cos x = (1 - \cos^2 x)/(1 + \cos x) = \sin^2 x / (1 + \cos x)

Techniques

  1. uu-substitution: sin⁡(2x)/x=2⋅sin⁡(2x)/(2x)→2\sin(2x)/x = 2 \cdot \sin(2x)/(2x) \to 2.
  2. Identities to rewrite in fundamental form.
  3. Conjugate: 1−cos⁡x=sin⁡2x/(1+cos⁡x)1 - \cos x = \sin^2 x / (1 + \cos x).
  4. Squeeze: −1≤sin⁡x≤1-1 \leq \sin x \leq 1, multiply by something →0\to 0.

Proof of lim⁡(1−cos⁡x)/x2=1/2\lim (1 - \cos x)/x^2 = 1/2

1−cos⁡xx2=2sin⁡2(x/2)x2=12⋅sin⁡2(x/2)(x/2)2→12⋅1=12\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}.

Proof of lim⁡tan⁡x/x=1\lim \tan x / x = 1

tan⁡x/x=(sin⁡x/x)⋅(1/cos⁡x)→1⋅1=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
  1. Ex. 48.1Application
    lim⁡x→0sin⁡(7x)/x\lim_{x \to 0} \sin(7x)/x. (Ans: 7.)
    Solve online
  2. Ex. 48.2Application
    lim⁡x→0sin⁡(3x)/sin⁡(5x)\lim_{x \to 0} \sin(3x)/\sin(5x). (Ans: 3/53/5.)
    Solve online
  3. Ex. 48.3Application
    lim⁡x→0tan⁡(2x)/sin⁡(3x)\lim_{x \to 0} \tan(2x)/\sin(3x). (Ans: 2/32/3.)
    Solve online
  4. Ex. 48.4Application
    lim⁡x→0sin⁡2x/x\lim_{x \to 0} \sin^2 x / x. (Ans: 0.)
    Solve online
  5. Ex. 48.5ApplicationAnswer key
    lim⁡x→0(1−cos⁡(2x))/x2\lim_{x \to 0} (1 - \cos(2x))/x^2. (Ans: 2.)
    Solve online
  6. Ex. 48.6Application
    lim⁡x→π/2(1−sin⁡x)/(π/2−x)2\lim_{x \to \pi/2} (1 - \sin x)/(\pi/2 - x)^2. (Ans: 1/21/2.)
    Solve online
  7. Ex. 48.7Application
    lim⁡x→0(tan⁡x−sin⁡x)/x3\lim_{x \to 0} (\tan x - \sin x)/x^3. (Ans: 1/21/2.)
    Solve online
  8. Ex. 48.8ApplicationAnswer key
    lim⁡x→0arcsin⁡(2x)/x\lim_{x \to 0} \arcsin(2x)/x. (Ans: 2.)
    Solve online
  9. Ex. 48.9Application
    lim⁡x→0arctan⁡(3x)/sin⁡(2x)\lim_{x \to 0} \arctan(3x)/\sin(2x). (Ans: 3/23/2.)
    Solve online
  10. Ex. 48.10Application
    lim⁡x→πsin⁡x/(x−π)\lim_{x \to \pi} \sin x/(x - \pi). (Ans: −1-1.)
    Solve online
  11. Ex. 48.11Application
    lim⁡x→0(cos⁡x−1)/(sin⁡x)\lim_{x \to 0} (\cos x - 1)/(\sin x). (Ans: 0.)
    Solve online
  12. Ex. 48.12Application
    lim⁡x→0x⋅cot⁡x\lim_{x \to 0} x \cdot \cot x. (Ans: 1.)
    Solve online
  13. Ex. 48.13Application
    lim⁡x→0sin⁡(x3)/x\lim_{x \to 0} \sin(x^3)/x. (Ans: 0.)
    Solve online
  14. Ex. 48.14ApplicationAnswer key
    lim⁡x→0sin⁡(x)sin⁡(2x)/x2\lim_{x \to 0} \sin(x)\sin(2x)/x^2. (Ans: 2.)
    Solve online
  15. Ex. 48.15ApplicationAnswer key
    lim⁡x→0(1−cos⁡(3x))/(1−cos⁡(2x))\lim_{x \to 0} (1 - \cos(3x))/(1 - \cos(2x)). (Ans: 9/49/4.)
    Solve online
  16. Ex. 48.16Application
    lim⁡x→0(1−cos⁡x)/x\lim_{x \to 0} (1 - \cos x)/x. (Ans: 0.)
    Solve online
  17. Ex. 48.17ApplicationAnswer key
    lim⁡x→0sin⁡(5x)⋅cot⁡(3x)\lim_{x \to 0} \sin(5x) \cdot \cot(3x). (Ans: 5/35/3.)
    Solve online
  18. Ex. 48.18ApplicationAnswer key
    lim⁡x→0(cos⁡x−cos⁡(3x))/x2\lim_{x \to 0} (\cos x - \cos(3x))/x^2. (Ans: 4.)
    Solve online
  19. Ex. 48.19Application
    lim⁡x→0sin⁡(πx)/sin⁡(π(1−x))\lim_{x \to 0} \sin(\pi x)/\sin(\pi(1-x)).
    Solve online
  20. Ex. 48.20Application
    lim⁡x→0sin⁡(sin⁡x)/x\lim_{x \to 0} \sin(\sin x)/x. (Ans: 1.)
    Solve online
  21. Ex. 48.21Application
    lim⁡x→π/4(tan⁡x−1)/(x−π/4)\lim_{x \to \pi/4} (\tan x - 1)/(x - \pi/4). (Ans: 2.)
    Solve online
  22. Ex. 48.22ApplicationAnswer key
    lim⁡x→π/2cos⁡x/(x−π/2)\lim_{x \to \pi/2} \cos x / (x - \pi/2). (Ans: −1-1.)
    Solve online
  23. Ex. 48.23Application
    lim⁡x→0(sec⁡x−1)/x2\lim_{x \to 0} (\sec x - 1)/x^2. (Ans: 1/21/2.)
    Solve online
  24. Ex. 48.24Application
    lim⁡x→0(tan⁡2x)/x2\lim_{x \to 0} (\tan^2 x)/x^2. (Ans: 1.)
    Solve online
  25. Ex. 48.25Application
    lim⁡x→0(1−cos⁡x⋅cos⁡(2x))/x2\lim_{x \to 0} (1 - \cos x \cdot \cos(2x))/x^2. (Ans: 5/25/2.)
    Solve online
  26. Ex. 48.26ApplicationAnswer key
    lim⁡x→0sin⁡(π−x)/x\lim_{x \to 0} \sin(\pi - x)/x.
    Solve online
  27. Ex. 48.27Application
    lim⁡x→0(sin⁡(a+x)−sin⁡a)/x\lim_{x \to 0} (\sin(a+x) - \sin a)/x. (Ans: cos⁡a\cos a.)
    Solve online
  28. Ex. 48.28Application
    lim⁡x→0(cos⁡(a+x)−cos⁡a)/x\lim_{x \to 0} (\cos(a+x) - \cos a)/x. (Ans: −sin⁡a-\sin a.)
    Solve online
  29. Ex. 48.29Application
    lim⁡x→0sin⁡x⋅ln⁡(1+x)/x2\lim_{x \to 0} \sin x \cdot \ln(1+x)/x^2. (Ans: 1.)
    Solve online
  30. Ex. 48.30Application
    lim⁡x→0(ex−cos⁡x)/x\lim_{x \to 0} (e^x - \cos x)/x. (Ans: 1.)
    Solve online
  31. Ex. 48.31ModelingAnswer key
    In control, H(s)H(s) with pole at jωj\omega — limit →∞\to \infty at resonance.
    Solve online
  32. Ex. 48.32Modeling
    Approximation sin⁡x≈x−x3/6\sin x \approx x - x^3/6 for small angles. Relative error at x=0.1x = 0.1?
    Solve online
  33. Ex. 48.33Modeling
    In mechanics, pendulum: θ¨=−(g/L)sin⁡θ≈−(g/L)θ\ddot \theta = -(g/L) \sin \theta \approx -(g/L)\theta for small. Justify via limit.
    Solve online
  34. Ex. 48.34Modeling
    Refraction: n1sin⁡θ1=n2sin⁡θ2n_1 \sin \theta_1 = n_2 \sin \theta_2. Paraxial approximation n1θ1≈n2θ2n_1 \theta_1 \approx n_2 \theta_2.
    Solve online
  35. Ex. 48.35Modeling
    In an AM signal, V(t)=(1+mcos⁡(ωmt))cos⁡(ωct)V(t) = (1 + m \cos(\omega_m t)) \cos(\omega_c t). For small mm, expand via limit.
    Solve online
  36. Ex. 48.36Modeling
    Single-slit diffraction: pattern ∝(sin⁡u/u)2\propto (\sin u/u)^2 with u=πasin⁡θ/λu = \pi a \sin\theta/\lambda. Compute limit at θ=0\theta = 0.
    Solve online
  37. Ex. 48.37Understanding
    Show lim⁡x→0sin⁡(kx)/x=k\lim_{x \to 0} \sin(kx)/x = k via variable change.
    Solve online
  38. Ex. 48.38Understanding
    Show (1−cos⁡x)/x2→1/2(1 - \cos x)/x^2 \to 1/2 using identity 1−cos⁡x=2sin⁡2(x/2)1 - \cos x = 2\sin^2(x/2).
    Solve online
  39. Ex. 48.39Challenge
    lim⁡x→0(sin⁡(sin⁡x)−x)/x3\lim_{x \to 0} (\sin(\sin x) - x)/x^3.
    Solve online
  40. Ex. 48.40ChallengeAnswer key
    lim⁡x→0(cos⁡(sin⁡x)−cos⁡x)/x4\lim_{x \to 0} (\cos(\sin x) - \cos x)/x^4.
    Solve online

Sources

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

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