lim_(x -- π/(4))((x-π/4)sin(3x-(3π)/(4)))/(2(1...

Question

lim_(x --> π/(4))((x-π/4)sin(3x-(3π)/(4)))/(2(1-sin 2x))=. . .
A. (3)/(2)
D. -(3)/(4)
B. (1)/(4)
E. -(3)/(2)
C. 0

G. Widosamodra · Accepted Answer

Jawaban : 3/4

Pembahasan :
Konsep :
lim_(x ➝ a) sin(x - a)/(x - a) = 1
sin x = cos (π/2 - x)
1 - cos 2x = 2 sin² x

lim_(x ➝ π/4) ((x-π/4)sin(3x-3π/4))/(2(1-sin 2x))
= ((π/4-π/4)sin(3(π/4)-3π/4))/(2(1-sin 2(π/4)))
= ((0)sin(3π/4-3π/4))/(2(1-sin π/2))
= 0/(2(1-1))
= 0/(2(0))
= 0/0 (tak tentu)

Karena hasilnya 0/0 (tak tentu) maka gunakan sifat limit trigonometri.
1 - sin 2x = 1 - cos (π/2 - 2x)
1 - sin 2x = 1 - cos 2(π/4 - x)
1 - sin 2x = 2 sin²(π/4 - x)

lim_(x ➝ π/4) ((x-π/4)sin(3x-3π/4)) / (2(1-sin 2x))
= lim_(x ➝ π/4) (-(π/4-x))sin(-3(π/4-x))) / (2(2 sin²(π/4 - x)))
= lim_(x ➝ π/4) (-(π/4-x))sin(-3(π/4-x))) / (4 sin²(π/4 - x))
= lim_(x ➝ π/4) (-1)(1/4) . (π/4-x)/sin(π/4 - x)  sin(-3(π/4-x))/sin (π/4 - x)
= -(1/4) . (1/1) . (-3/1)
= 3/4

Jadi, lim_(x ➝ π/4) ((x-π/4)sin(3x-3π/4))/(2(1-sin 2x)) = 3/4

lim_(x --> π/(4))((x-π/4)sin(3x-(3π)/(4)))/(2(1-sin 2x))=. . . A. (3)/(2) D. -(3)/(4) B. (1)/(4) E. -(3)/(2) C. 0

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