tentukan nilai limit X mendekati 0 cos 2x - cos 4x/1 - cos 6x

Nilai limit x mendekati 0 cos 2x – cos 4x/1 – cos 6x adalah ⅓. Rumus limit trigonometri

• [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{sin \: ax}{bx} = \lim \limits_{{x}{\rightarrow}{0}} \frac{ax}{ sin \: bx} = \frac{a}{b} [/tex]
• [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{tan \: ax}{bx} = \lim \limits_{{x}{\rightarrow}{0}} \frac{ax}{ tan \: bx} = \frac{a}{b} [/tex]

• [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{sin \: ax}{sin \: bx} = \lim \limits_{{x}{\rightarrow}{0}} \frac{tan \: ax}{tan \: bx} = \frac{a}{b} [/tex]  
• [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{sin \: ax}{tan \: bx} = \lim \limits_{{x}{\rightarrow}{0}} \frac{tan \: ax}{sin \: bx} = \frac{a}{b} [/tex]  

Jika berbentuk cosinus maka kita ubah dulu menjadi

• cos² ax = 1 – sin² ax
• cos ax = 1 – 2 sin² ½ ax
• cos A – cos B = –2 sin ½ (A + B) sin ½ (A – B)

Pembahasan

[tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{cos \: 2x - cos \: 4x}{1 - cos \: 6x} [/tex]

= [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{-2 \: sin \: \frac{1}{2} (2x + 4x) \: sin \: \frac{1}{2} (2x - 4x)}{1 - (1 - 2 \: sin^{2} \: \frac{1}{2}(6x)} [/tex]

= [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{-2 \: sin \: \frac{1}{2} (6x) \: sin \: \frac{1}{2} (-2x)}{2 \: sin^{2} \: 3x} [/tex]

= [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{-2 \: sin \: 3x \: sin \: (-x)}{2 \: sin \: 3x \: sin \: 3x} [/tex]

= [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{-2}{2} \: . \: \frac{sin \: 3x}{sin \: 3x} \: . \: \frac{sin \: (-x)}{sin \: 3x} [/tex]

= –1 . 1 . –⅓

= ⅓

Pelajari lebih lanjut  

Contoh soal lain tentang  limit trigonometri

• Lim (x tan x)/(2 cos² x – 2): brainly.co.id/tugas/8875767
• Lim (sin 2x)/(sin 6x): brainly.co.id/tugas/1778468
• Lim (x² + sin² 3x)/(2 tan 2x²): brainly.co.id/tugas/10096707

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Detil Jawaban    

Kelas : 12

Mapel : Matematika Peminatan

Kategori : Limit Trigonometri dan Limit Tak Hingga

Kode : 12.2.1

Kata Kunci : Limit trigonometri