x→−2lim​4x2+8x(x3+8)(x2+4x+3)​=...

Pertanyaan

limit as x rightwards arrow negative 2 of fraction numerator left parenthesis x cubed plus 8 right parenthesis left parenthesis x squared plus 4 x plus 3 right parenthesis over denominator 4 x squared plus 8 x end fraction equals... 

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

hasil dari limit as x rightwards arrow negative 2 of fraction numerator left parenthesis x cubed plus 8 right parenthesis left parenthesis x squared plus 4 x plus 3 right parenthesis over denominator 4 x squared plus 8 x end fraction adalah 3 over 2.

Pembahasan

Pembahasan

Substitusi terlebih dahulu:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow negative 2 of fraction numerator open parentheses x cubed plus 8 close parentheses open parentheses x squared plus 4 x plus 3 close parentheses over denominator 4 x squared plus 8 x end fraction end cell equals cell fraction numerator open parentheses left parenthesis negative 2 right parenthesis cubed plus 8 close parentheses open parentheses left parenthesis negative 2 right parenthesis squared plus 4 left parenthesis negative 2 right parenthesis plus 3 close parentheses over denominator 4 left parenthesis negative 2 right parenthesis squared plus 8 left parenthesis negative 2 right parenthesis end fraction end cell row blank equals cell fraction numerator open parentheses negative 8 plus 8 close parentheses open parentheses 4 minus 8 plus 3 close parentheses over denominator 4 times 4 minus 16 end fraction end cell row blank equals cell fraction numerator open parentheses 0 close parentheses open parentheses negative 1 close parentheses over denominator 16 minus 16 end fraction end cell row blank equals cell fraction numerator open parentheses 0 close parentheses open parentheses negative 1 close parentheses over denominator 0 end fraction end cell row blank equals cell 0 over 0 end cell end table

Karena 0 over 0 bentuk tak tentu, maka mesti diselesaikan dengan pemfaktoran dan bisa dipisah dengan sifat limit sebagai berikut:

Ingat: open parentheses a cubed plus b cubed close parentheses equals open parentheses a plus b close parentheses open parentheses a squared minus a b plus b squared close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow negative 2 of fraction numerator open parentheses x cubed plus 8 close parentheses open parentheses x squared plus 4 x plus 3 close parentheses over denominator 4 x squared plus 8 x end fraction end cell equals cell limit as x rightwards arrow negative 2 of fraction numerator open parentheses x cubed plus 2 cubed close parentheses over denominator 4 x squared plus 8 x end fraction times limit as x rightwards arrow negative 2 of open parentheses x squared plus 4 x plus 3 close parentheses end cell row blank equals cell limit as x rightwards arrow negative 2 of fraction numerator left parenthesis x plus 2 right parenthesis open parentheses x squared minus x times 2 plus 2 squared close parentheses over denominator open parentheses 4 x close parentheses open parentheses x plus 2 close parentheses end fraction times open parentheses left parenthesis negative 2 right parenthesis squared plus 4 left parenthesis negative 2 right parenthesis plus 3 close parentheses end cell row blank equals cell limit as x rightwards arrow negative 2 of fraction numerator open parentheses x squared minus 2 x plus 4 close parentheses over denominator 4 x end fraction times open parentheses 4 minus 8 plus 3 close parentheses end cell row blank equals cell fraction numerator open parentheses left parenthesis negative 2 right parenthesis squared minus 2 left parenthesis negative 2 right parenthesis plus 4 close parentheses over denominator 4 left parenthesis negative 2 right parenthesis end fraction times open parentheses negative 1 close parentheses end cell row blank equals cell fraction numerator open parentheses 4 plus 4 plus 4 close parentheses over denominator negative 8 end fraction times open parentheses negative 1 close parentheses end cell row blank equals cell fraction numerator 12 over denominator negative 8 end fraction times open parentheses negative 1 close parentheses end cell row blank equals cell 3 over 2 end cell end table

Jadi hasil dari limit as x rightwards arrow negative 2 of fraction numerator left parenthesis x cubed plus 8 right parenthesis left parenthesis x squared plus 4 x plus 3 right parenthesis over denominator 4 x squared plus 8 x end fraction adalah 3 over 2.

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