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x → 4 1 ​ π lim ​ x − 4 1 ​ π tan x − cot x ​ = ...

       

  1. infinity    

  2. 4  

  3. 2    

  4. 1  

  5. 0   

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S. Nur

Master Teacher

Jawaban terverifikasi

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Pembahasan

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Ingat kembali beberapa sifat berikut. Dari aturan di atas, maka diperoleh Misalkan . Jika maka Sehingga diperoleh Dengan demikian, . Jadi, jawaban yang benar adalah B.

Ingat kembali  beberapa sifat berikut.

  • tan space x equals fraction numerator sin space x over denominator cos space x end fraction space dan space c o t space x equals fraction numerator cos space x over denominator sin space x end fraction 
     
  • cos space 2 x equals cos squared x minus sin squared x    
     
  • sin space 2 x equals 2 times sin space x times cos space x 
     
  • limit as x rightwards arrow c of k f left parenthesis x right parenthesis equals k limit as x rightwards arrow c of f left parenthesis x right parenthesis   
     
  • cot space left parenthesis straight pi over 2 plus x right parenthesis equals negative tan space x 
     
  • limit as x rightwards arrow 0 of fraction numerator tan space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator tan space b x end fraction equals a over b    

Dari aturan di atas, maka diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator tan space x minus cot space x over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator sin space x over denominator cos space x end fraction end style minus begin display style fraction numerator cos space x over denominator sin space x end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator sin squared space x minus cos squared space x over denominator sin space x space cos space x end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator negative cos space 2 x over denominator begin display style 1 half end style times 2 times sin space x space cos space x end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator negative cos space 2 x over denominator begin display style 1 half times sin space 2 x end style end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style negative 2 times fraction numerator begin display style cos space 2 x end style over denominator begin display style sin space 2 x end style end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style negative 2 times cot space 2 x end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell end table 

Misalkan y equals x minus 1 fourth straight pi.

Jika x rightwards arrow 1 fourth pi maka

 table attributes columnalign right center left columnspacing 0px end attributes row y rightwards arrow cell 1 fourth straight pi minus 1 fourth straight pi end cell row straight y rightwards arrow 0 end table           

Sehingga diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator negative 2 space cot space 2 x over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space cot space 2 left parenthesis y plus begin display style 1 fourth end style straight pi right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space cot space left parenthesis 2 y plus begin display style 1 half end style straight pi right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space cot space left parenthesis begin display style 1 half end style straight pi plus 2 straight y right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space left parenthesis negative tan space left parenthesis 2 straight y right parenthesis right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator 2 space tan space left parenthesis 2 straight y right parenthesis over denominator straight y end fraction end cell row blank equals cell 2 space limit as y rightwards arrow 0 of space fraction numerator tan space 2 y over denominator y end fraction end cell row blank equals cell 2 times 2 end cell row blank equals 4 end table 

Dengan demikian, limit as x rightwards arrow 1 fourth straight pi of fraction numerator tan space x minus cot space x over denominator x minus begin display style 1 fourth end style straight pi end fraction equals 4.

Jadi, jawaban yang benar adalah B.

 

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