Pertanyaan

Nilai x → ∞ lim ( 3 x + 1 ) ( x 3 − 2 ) − 3 x 4 ( 2 x − 3 ) 2 ( 2 − x ) = ...

Nilai $x \to \infty lim \frac{( 2 x - 3 ) ^{2} ( 2 - x )}{( 3 x + 1 ) ( x ^{3} - 2 ) - 3 x ^{4}} =$ ...

R. Febrianti

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

$begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator open parentheses 2 x minus 3 close parentheses squared open parentheses 2 minus x close parentheses over denominator open parentheses 3 x plus 1 close parentheses open parentheses x cubed minus 2 close parentheses minus 3 x to the power of 4 end fraction equals negative 4 end style$

Pembahasan

Uraikan limit tersebut, kemudian gunakan sifat limit tak hingga bagilah dengan variabel pangkat tertinggi kemudian ambilah nilai limitnya seperti berikut: x → ∞ lim ( 3 x + 1 ) ( x 3 − 2 ) − 3 x 4 ( 2 x − 3 ) 2 ( 2 − x ) = x → ∞ lim ( 3 x 4 − 6 x + x 3 − 2 ) − 3 x 4 ( 4 x 2 − 12 x + 9 ) ( 2 − x ) = x → ∞ lim − 3 x 4 + 3 x 4 − 6 x + x 3 − 2 8 x 2 − 4 x 3 − 24 x + 12 x 2 + 18 − 9 x = x → ∞ lim x 3 − 6 x − 2 − 4 x 3 + 20 x 2 − 33 x + 18 = x → ∞ lim x 3 x 3 − x 3 6 x − x 3 2 x 3 − 4 x 3 + x 3 20 x 2 − x 3 33 x + x 3 18 = x → ∞ lim 1 − x 2 6 − x 3 2 − 4 + x 20 − x 2 33 + x 3 18 = 1 − 0 − 0 − 4 + 0 − 0 + 0 = 1 − 4 = − 4 Dengan demikian,

Uraikan limit tersebut, kemudian gunakan sifat limit tak hingga bagilah dengan variabel pangkat tertinggi kemudian ambilah nilai limitnya seperti berikut:

$x \to \infty lim \frac{( 2 x - 3 ) ^{2} ( 2 - x )}{( 3 x + 1 ) ( x ^{3} - 2 ) - 3 x ^{4}} = x \to \infty lim \frac{( 4 x ^{2} - 12 x + 9 ) ( 2 - x )}{( 3 x ^{4} - 6 x + x ^{3} - 2 ) - 3 x ^{4}} = x \to \infty lim \frac{8 x ^{2} - 4 x ^{3} - 24 x + 12 x ^{2} + 18 - 9 x}{- 3 x ^{4} + 3 x ^{4} - 6 x + x ^{3} - 2} = x \to \infty lim \frac{- 4 x ^{3} + 20 x ^{2} - 33 x + 18}{x ^{3} - 6 x - 2} = x \to \infty lim \frac{\frac{- 4 x ^{3}}{x ^{3}} + \frac{20 x ^{2}}{x ^{3}} - \frac{33 x}{x ^{3}} + \frac{18}{x ^{3}}}{\frac{x ^{3}}{x ^{3}} - \frac{6 x}{x ^{3}} - \frac{2}{x ^{3}}} = x \to \infty lim \frac{- 4 + \frac{20}{x} - \frac{33}{x ^{2}} + \frac{18}{x ^{3}}}{1 - \frac{6}{x ^{2}} - \frac{2}{x ^{3}}} = \frac{- 4 + 0 - 0 + 0}{1 - 0 - 0} = \frac{- 4}{1} = - 4$

Dengan demikian, $begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator open parentheses 2 x minus 3 close parentheses squared open parentheses 2 minus x close parentheses over denominator open parentheses 3 x plus 1 close parentheses open parentheses x cubed minus 2 close parentheses minus 3 x to the power of 4 end fraction equals negative 4 end style$