RoboguruRoboguru
SD

x→41​πlim​x−41​πtan x−cot x​=...

Pertanyaan

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...       

  1. infinity    

  2. 4  

  3. 2    

  4. 1  

  5. 0   

N. Puspita

Master Teacher

Jawaban terverifikasi

Pembahasan

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.

 

68

5.0 (1 rating)

Pertanyaan serupa

Nilai x→0lim​sin 4x⋅tan x3x⋅tan 2x⋅cos x​=....

3

0.0

Jawaban terverifikasi

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Roboguru

Coba GRATIS Aplikasi Ruangguru

Download di Google PlayDownload di AppstoreDownload di App Gallery

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ruangguru WhatsApp

081578200000

Email info@ruangguru.com

info@ruangguru.com

Contact 02140008000

02140008000

Ikuti Kami

©2022 Ruangguru. All Rights Reserved PT. Ruang Raya Indonesia