Pertanyaan

x → ∞ lim 2 x 5 + 3 x 3 − 2 x − 1 ( x − 2 ) 2 ( x 3 + 5 )

$x \to \infty lim \frac{( x - 2 ) ^{2} ( x ^{3} + 5 )}{2 x ^{5} + 3 x ^{3} - 2 x - 1}$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

Pembahasan

Ingat: Jadi

Ingat: limit as x rightwards arrow infinity of k over x to the power of n equals k over infinity to the power of n equals 0

$limit as x rightwards arrow infinity of fraction numerator left parenthesis x minus 2 right parenthesis squared left parenthesis x cubed plus 5 right parenthesis over denominator 2 x to the power of 5 plus 3 x cubed minus 2 x minus 1 end fraction equals limit as x rightwards arrow infinity of fraction numerator left parenthesis x squared minus 4 x plus 4 right parenthesis left parenthesis x cubed plus 5 right parenthesis over denominator 2 x to the power of 5 plus 3 x cubed minus 2 x minus 1 end fraction equals limit as x rightwards arrow infinity of fraction numerator x to the power of 5 plus 5 x squared minus 4 x to the power of 4 minus 20 x plus 4 x cubed plus 20 over denominator 2 x to the power of 5 plus 3 x cubed minus 2 x minus 1 end fraction cross times fraction numerator begin display style 1 over x to the power of 5 end style over denominator begin display style 1 over x to the power of 5 end style end fraction equals limit as x rightwards arrow infinity of fraction numerator begin display style x to the power of 5 over x to the power of 5 minus fraction numerator 4 x to the power of 4 over denominator x to the power of 5 end fraction plus fraction numerator 4 x cubed over denominator x to the power of 5 end fraction end style plus begin display style fraction numerator 5 x squared over denominator x to the power of 5 end fraction end style minus begin display style fraction numerator 20 x over denominator x to the power of 5 end fraction end style plus begin display style 20 over x to the power of 5 end style over denominator begin display style fraction numerator 2 x to the power of 5 over denominator x to the power of 5 end fraction end style plus begin display style fraction numerator 3 x cubed over denominator x to the power of 5 end fraction end style minus begin display style fraction numerator 2 x over denominator x to the power of 5 end fraction end style minus begin display style 1 over x to the power of 5 end style end fraction equals limit as x rightwards arrow infinity of fraction numerator begin display style 1 minus 4 over x plus 4 over x squared plus 5 over x cubed minus 20 over x to the power of 4 plus 20 over x to the power of 5 end style over denominator begin display style 2 plus 3 over x squared minus 2 over x to the power of 4 minus 1 over x to the power of 5 end style end fraction equals fraction numerator begin display style 1 minus 4 over infinity plus 4 over infinity squared plus 5 over infinity cubed minus 20 over infinity to the power of 4 plus 20 over infinity to the power of 5 end style over denominator begin display style 2 plus 3 over infinity squared minus 2 over infinity to the power of 4 minus 1 over infinity to the power of 5 end style end fraction equals fraction numerator 1 plus begin display style 0 end style minus begin display style 0 end style minus 0 plus begin display style 0 end style plus begin display style 0 end style over denominator 2 plus begin display style 0 end style minus begin display style 0 end style minus begin display style 0 end style end fraction equals 1 half$

Jadi $limit as x rightwards arrow infinity of fraction numerator left parenthesis x minus 2 right parenthesis squared left parenthesis x cubed plus 5 right parenthesis over denominator 2 x to the power of 5 plus 3 x cubed minus 2 x minus 1 end fraction equals 1 half$