Pertanyaan

Nilai lim x → ∞ ( x − x 2 − 5 x ) = ....

Nilai $lim_{x \to \infty} (x - x^{2} - 5 x) = ....$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Langkah-langkah limit tak hingga bentuk akar: Kali dengan akar sekawan. Bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut. Substitusi x mendekati Langkah pertama (kali dengan akar sekawan): Ingat: . Langkah kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut): Langkah ketiga (substitusi x mendekati ): Ingat: . Jadi . Oleh karena itu, jawaban yang tepat adalah D.

Langkah-langkah limit tak hingga bentuk akar:

Kali dengan akar sekawan.
Bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut.
Substitusi x mendekati

Langkah pertama (kali dengan akar sekawan):

Ingat: left parenthesis a minus b right parenthesis left parenthesis a plus b right parenthesis equals a squared minus b squared .

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of open parentheses x minus square root of x squared minus 5 x end root close parentheses end cell equals cell limit as x rightwards arrow infinity of open parentheses x minus square root of x squared minus 5 x end root close parentheses cross times fraction numerator open parentheses x plus square root of x squared minus 5 x end root close parentheses over denominator open parentheses x plus square root of x squared minus 5 x end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator open parentheses x squared minus open parentheses square root of x squared minus 5 x end root close parentheses squared close parentheses over denominator open parentheses x plus square root of x squared minus 5 x end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator open parentheses x squared minus x squared plus 5 x close parentheses over denominator open parentheses x plus square root of x squared minus 5 x end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator 5 x over denominator x plus square root of x squared minus 5 x end root end fraction end cell end table$

Langkah kedua (bagi dengan suku berpangkat tertinggi dari pembilang atau penyebut):

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

Langkah ketiga (substitusi x mendekati infinity ):

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 .

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

Jadi table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x minus square root of x squared minus 5 x end root close parentheses end cell end table equals 2 comma 5 .

Oleh karena itu, jawaban yang tepat adalah D.

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