Limit x=0 dari x^2√4-x/cosx-cos3x

Nilai limit x = 0 dari x^2 √4 – x /cos x – cos 3x adalah ½. 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{x^{2} \: \sqrt{4 - x}}{cos \: x - cos \: 3x}[/tex]

= [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{ x^{2} \: \sqrt{4 - x}}{-2 \: sin \: \frac{1}{2} (x + 3x) \: sin \: \frac{1}{2} (x - 3x)}[/tex]

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

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

= [tex] \lim \limits_{{x}{\rightarrow}{0}} \frac{\sqrt{4 - x}}{-2} \: . \: \frac{x}{sin \: 2x} \: . \: \frac{x}{sin \: (-x)} [/tex]

= [tex]\frac{\sqrt{4 - 0}}{-2} \: . \: \frac{1}{2} \: . \: \frac{1}{-1} [/tex]

= [tex]\frac{\sqrt{4}}{-2} \: . \: \frac{1}{2} \: . \: -1 [/tex]

= [tex]\frac{2}{-2} \: . \: \frac{1}{2} \: . \: -1 [/tex]

= [tex]-1 \: . \: \frac{1}{2} \: . \: -1 [/tex]

= [tex]\frac{1}{2}[/tex]

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 : 10.2.1

Kata Kunci : Limit trigonometri