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x→∞lim​((x+p)(x+q)​−x)=...

Pertanyaan

limit as straight x rightwards arrow infinity of open parentheses square root of open parentheses x plus p close parentheses open parentheses x plus q close parentheses end root minus x close parentheses equals...  

  1. begin mathsize 14px style 0 end style 

  2. p q 

  3. p minus q   

  4. 1 half open parentheses p plus q close parentheses   

  5. p plus q 

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Langkah-langkah limit tak hingga bentuk akar:

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

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 blank blank cell limit as straight x rightwards arrow infinity of open parentheses square root of open parentheses x plus p close parentheses open parentheses x plus q close parentheses end root minus x close parentheses end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow infinity of open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root minus x close parentheses cross times fraction numerator open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses over denominator open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow infinity of end cell end table fraction numerator table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root close parentheses squared minus x squared close parentheses end cell end table over denominator table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses end cell end table end fraction equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator open parentheses left parenthesis x squared plus p plus q right parenthesis x plus p q minus x squared close parentheses over denominator open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses end fraction end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator open parentheses left parenthesis p plus q right parenthesis x plus p q close parentheses over denominator open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses 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 straight x rightwards arrow infinity of fraction numerator open parentheses left parenthesis p plus q right parenthesis x plus p q close parentheses over denominator open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses end fraction end cell equals cell limit as straight x rightwards arrow infinity of fraction numerator open parentheses left parenthesis p plus q right parenthesis x plus p q close parentheses over denominator open parentheses square root of x squared plus left parenthesis p plus q right parenthesis x plus p q end root plus x close parentheses 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 straight x rightwards arrow infinity of fraction numerator open parentheses begin display style fraction numerator left parenthesis p plus q right parenthesis x over denominator x end fraction end style plus begin display style fraction numerator p q over denominator x end fraction end style close parentheses over denominator open parentheses square root of begin display style x squared over x squared end style plus begin display style fraction numerator left parenthesis p plus q right parenthesis x over denominator x squared end fraction end style plus begin display style fraction numerator p q over denominator x squared end fraction end style end root plus begin display style x over x end style close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow infinity of fraction numerator open parentheses begin display style left parenthesis p plus q right parenthesis plus fraction numerator p q over denominator x end fraction end style close parentheses over denominator open parentheses square root of begin display style 1 plus fraction numerator left parenthesis p plus q right parenthesis over denominator x end fraction plus fraction numerator p q over denominator x squared end fraction end style end root plus begin display style 1 end style close parentheses 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 straight x rightwards arrow infinity of fraction numerator open parentheses begin display style left parenthesis p plus q right parenthesis plus fraction numerator p q over denominator x end fraction end style close parentheses over denominator open parentheses square root of begin display style 1 plus fraction numerator left parenthesis p plus q right parenthesis over denominator x end fraction plus fraction numerator p q over denominator x squared end fraction end style end root plus begin display style 1 end style close parentheses end fraction end cell equals cell fraction numerator open parentheses begin display style left parenthesis p plus q right parenthesis plus fraction numerator p q over denominator infinity end fraction end style close parentheses over denominator open parentheses square root of begin display style 1 plus fraction numerator left parenthesis p plus q right parenthesis over denominator infinity end fraction plus fraction numerator p q over denominator infinity squared end fraction end style end root plus begin display style 1 end style close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses begin display style left parenthesis p plus q right parenthesis plus 0 end style close parentheses over denominator open parentheses square root of begin display style 1 plus 0 plus 0 end style end root plus begin display style 1 end style close parentheses end fraction end cell row blank equals cell fraction numerator p plus q over denominator 1 plus 1 end fraction end cell row blank equals cell 1 half left parenthesis p plus q right parenthesis end cell end table

Jadi, hasil dari limit as x rightwards arrow infinity of open parentheses square root of open parentheses x plus p close parentheses open parentheses x plus q close parentheses end root minus x close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses p plus q close parentheses end cell end table.

Oleh karena itu, jawaban yang benar adalah D.

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