lim x->1phi/4 ((x-phi/4)tan(3x-3phi/4)/(2(1-sin2x))

3/4

Pembahasan

Untuk menyelesaikan limit ini, kita dapat menggunakan aturan L'Hopital karena ketika x mendekati π/4, baik pembilang maupun penyebut akan bernilai 0.

Limit = lim x->π/4 (x - π/4)tan(3x - 3π/4) / (2(1 - sin(2x)))

Turunan dari pembilang:
d/dx [(x - π/4)tan(3x - 3π/4)]
= 1 * tan(3x - 3π/4) + (x - π/4) * sec^2(3x - 3π/4) * 3
= tan(3x - 3π/4) + 3(x - π/4)sec^2(3x - 3π/4)

Turunan dari penyebut:
d/dx [2(1 - sin(2x))]
= 2 * (-cos(2x) * 2)
= -4cos(2x)

Sekarang, terapkan aturan L'Hopital:
Limit = lim x->π/4 [tan(3x - 3π/4) + 3(x - π/4)sec^2(3x - 3π/4)] / [-4cos(2x)]

Ganti x dengan π/4:
tan(3(π/4) - 3π/4) = tan(0) = 0
3(π/4 - π/4)sec^2(3(π/4) - 3π/4) = 3(0)sec^2(0) = 0
-4cos(2(π/4)) = -4cos(π/2) = -4 * 0 = 0

Karena masih berbentuk 0/0, kita terapkan aturan L'Hopital lagi.

Turunan dari pembilang yang baru:
d/dx [tan(3x - 3π/4) + 3(x - π/4)sec^2(3x - 3π/4)]
= sec^2(3x - 3π/4) * 3 + [3sec^2(3x - 3π/4) + 3(x - π/4) * 2sec(3x - 3π/4) * sec(3x - 3π/4)tan(3x - 3π/4) * 3]
= 3sec^2(3x - 3π/4) + 3sec^2(3x - 3π/4) + 18(x - π/4)sec^2(3x - 3π/4)tan(3x - 3π/4)
= 6sec^2(3x - 3π/4) + 18(x - π/4)sec^2(3x - 3π/4)tan(3x - 3π/4)

Turunan dari penyebut yang baru:
d/dx [-4cos(2x)]
= -4 * (-sin(2x) * 2)
= 8sin(2x)

Sekarang, terapkan aturan L'Hopital lagi:
Limit = lim x->π/4 [6sec^2(3x - 3π/4) + 18(x - π/4)sec^2(3x - 3π/4)tan(3x - 3π/4)] / [8sin(2x)]

Ganti x dengan π/4:
6sec^2(0) + 18(0)sec^2(0)tan(0) = 6(1)^2 + 0 = 6
8sin(π/2) = 8(1) = 8

Limit = 6 / 8 = 3 / 4