Nilai limit x mendekati phi/4 1-tanx/sinx-cosx

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Nilai limit x mendekati phi/4 1-tanx/sinx-cosx

Jawaban Terkonfirmasi

Nilai limit x mendekati phi/4 1 – tan x / sin x – cos x adalah - sqrt{2}. Definisi: lim limits_{{x}{rightarrow}{a}} f(x) = f(a) dengan f(a) ≠ frac{0}{0} ≠  frac{infty}{infty} ≠ ∞ – ∞

Rumus limit trigonometri

  • lim limits_{{x}{rightarrow}{0}} frac{sin : ax}{bx} = lim limits_{{x}{rightarrow}{0}} frac{ax}{ sin : bx} = frac{a}{b}
  • lim limits_{{x}{rightarrow}{0}} frac{tan : ax}{bx} = lim limits_{{x}{rightarrow}{0}} frac{ax}{ tan : bx} = frac{a}{b}
  • lim limits_{{x}{rightarrow}{0}} frac{sin : ax}{sin : bx} = lim limits_{{x}{rightarrow}{0}} frac{tan : ax}{tan : bx} = frac{a}{b}  
  • lim limits_{{x}{rightarrow}{0}} frac{sin : ax}{tan : bx} = lim limits_{{x}{rightarrow}{0}} frac{tan : ax}{sin : bx} = frac{a}{b}  

Jika berbentuk cosinus maka kita ubah dulu menjadi

  • cos² ax = 1 – sin² ax
  • cos ax = 1 – 2 sin² ½ ax
  • cos x – cos y = –2 sin ½ (x + y) sin ½ (x – y)

Pembahasan

lim limits_{{x}{rightarrow}{frac{pi}{4}}} frac{1 - tan : x}{sin : x - cos : x}

= lim limits_{{x}{rightarrow}{frac{pi}{4}}} frac{1}{sin : x - cos : x} : . : (1 - tan : x)

= lim limits_{{x}{rightarrow}{frac{pi}{4}}} frac{1}{sin : x - cos : x} : . : (1 - frac{sin : x}{cos : x})

= lim limits_{{x}{rightarrow}{frac{pi}{4}}} frac{1}{sin : x - cos : x} : . : (frac{cos : x - sin : x}{cos : x})

= lim limits_{{x}{rightarrow}{frac{pi}{4}}} frac{1}{sin : x - cos : x} : . : (frac{-1(sin : x - cos : x)}{cos : x})

= lim limits_{{x}{rightarrow}{frac{pi}{4}}} (frac{-1}{cos : x})

= frac{-1}{cos : frac{pi}{4}}

= frac{-1}{frac{1}{2} sqrt{2}}

= frac{-1}{frac{1}{2} sqrt{2}} : . : frac{sqrt{2}}{sqrt{2}}

= frac{- sqrt{2}}{1}

= - sqrt{2}

Pelajari lebih lanjut  

Contoh soal lain tentang limit trigonometri

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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