Pertanyaan

Tentukan nilai limit tak hingga berikut! x → ∞ lim 5 x 3 + 2 x 2 + 1 5 + 2 x + x 3

Tentukan nilai limit tak hingga berikut!

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

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

.

$begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator 5 plus 2 x plus x cubed over denominator 5 x cubed plus 2 x squared plus 1 end fraction equals 1 fifth end style$ .

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 .

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 5 plus 2 x plus x cubed over denominator 5 x cubed plus 2 x squared plus 1 end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator 5 plus 2 x plus x cubed over denominator 5 x cubed plus 2 x squared plus 1 end fraction times fraction numerator begin display style 1 over x cubed end style over denominator begin display style 1 over x cubed end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 5 over x cubed end style plus begin display style fraction numerator 2 x over denominator x cubed end fraction end style plus begin display style x cubed over x cubed end style over denominator begin display style fraction numerator 5 x cubed over denominator x cubed end fraction end style plus begin display style fraction numerator 2 x squared over denominator x cubed end fraction end style plus begin display style 1 over x cubed end style end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator begin display style 5 over x cubed end style plus begin display style 2 over x squared end style plus 1 over denominator 5 plus begin display style 2 over x end style plus begin display style 1 over x cubed end style end fraction end cell row blank equals cell fraction numerator begin display style 5 over infinity cubed plus 2 over infinity squared plus 1 end style over denominator 5 plus begin display style 2 over infinity end style plus begin display style 1 over infinity cubed end style end fraction end cell row blank equals cell fraction numerator 0 plus 0 plus 1 over denominator 5 plus 0 plus 0 end fraction end cell row blank equals cell 1 fifth end cell end table$

Jadi $begin mathsize 14px style limit as x rightwards arrow infinity of fraction numerator 5 plus 2 x plus x cubed over denominator 5 x cubed plus 2 x squared plus 1 end fraction equals 1 fifth end style$ .