lim_(x→3) (4x-6-√(x²+3x+18))/(3-x)=.......

Question

lim_(x→3) (4x-6-√(x²+3x+18))/(3-x)=....

E. Hayuning · Accepted Answer

Jawaban yang benar adalah -13/4.

Pembahasan :
Konsep :
Jika Lim x→a f(x)/g(x) = 0/0 maka gunakan cara lain.
(√a - √b)(√a + √b) = a - b
(a - b) = √(a - b)²
(a - b) = -1(b - a)

Lim x→3 (4x - 6 - √(x² + 3x + 18))/(3 - x)
= (4(3) - 6 - √((3)² + 3(3) + 18))/(3 - 3)
= (12 - 6 - √(9 + 9 + 18))/(0)
= (6 - √36)/(0)
= (6 - 6)/(0)
= 0/0 tak tentu

Karena hasilnya 0/0 maka gunakan perkalian sekawan.

Lim x→3 (4x - 6 - √(x² + 3x + 18))/(3 - x)
= Lim x→3 ((4x - 6) - √(x² + 3x + 18))/(3 - x)
= Lim x→3 (√(4x - 6)² - √(x² + 3x + 18))/(3 - x)
= Lim x→3 (√(16x² - 48x + 36) - √(x² + 3x + 18))/(3 - x)
= Lim x→3 (√(16x² - 48x + 36) - √(x² + 3x + 18))/(3 - x)  .   (√(16x² - 48x + 36) + √(x² + 3x + 18))/(√(16x² - 48x + 36) + √(x² + 3x + 18))
= Lim x→3 ((16x² - 48x + 36) - (x² + 3x + 18)) / (3 - x)(√(16x² - 48x + 36) + √(x² + 3x + 18))
= Lim x→3 (16x² - 48x + 36 - x² - 3x - 18) / (3 - x)(√(4x - 6)² + √(x² + 3x + 18))
= Lim x→3 (15x² - 51x + 18) / (3 - x)((4x - 6) + √(x² + 3x + 18))
= Lim x→3 3(5x² - 17x + 6) / (3 - x)((4x - 6) + √(x² + 3x + 18))
= Lim x→3 3(5x - 2)(x - 3) / (3 - x)((4x - 6) + √(x² + 3x + 18))
= Lim x→3 3(5x - 2)(-1)(3 - x) / (3 - x)((4x - 6) + √(x² + 3x + 18))
= Lim x→3 -3(5x - 2)/((4x - 6) + √(x² + 3x + 18))
= -3(5(3) - 2)/((4(3) - 6) + √((3)² + 3(3) + 18))
= -3(15 - 2)/((12 - 6) + √(9 + 9 + 18))
= -3(13)/(6 + √36)
= -3(13)/(6 + 6)
= -3(13)/12
= -13/4

Jadi, Lim x→3 (4x - 6 - √(x² + 3x + 18))/(3 - x) = -13/4
Semoga membantu ya.

lim_(x→3) (4x-6-√(x²+3x+18))/(3-x)=....

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