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x→31​πlim​π−3x1−2cosx​=…

Pertanyaan

limit as x rightwards arrow 1 third straight pi of fraction numerator 1 minus 2 cos space x over denominator straight pi minus 3 straight x end fraction equals horizontal ellipsis 

  1. negative square root of 3 

  2. negative 1 third square root of 3 

  3. 0 

  4. 1 third square root of 3 

  5. square root of 3 

Pembahasan Soal:

Ingat kembali pemfaktoran, identitas trigonometri, selisih dua sudut pada sin dan limit fungsi trigonometri berikut ini.

  • left parenthesis a plus b right parenthesis left parenthesis a minus b right parenthesis equals a squared minus b squared

 

  • sin squared x plus cos squared x equals 1 

 

  • sin space A space cos space B minus cos space A space sin space B equals sin left parenthesis A minus B right parenthesis

 

  •  limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus c right parenthesis over denominator b left parenthesis x minus c right parenthesis end fraction equals limit as x rightwards arrow c of fraction numerator a left parenthesis x minus c right parenthesis over denominator sin space b left parenthesis x minus c right parenthesis end fraction equals a over b

 

  • limit as x rightwards arrow c of f left parenthesis x right parenthesis times g left parenthesis x right parenthesis equals limit as x rightwards arrow c of space f left parenthesis x right parenthesis times limit as x rightwards arrow c of space g left parenthesis x right parenthesis

 

  • limit as x rightwards arrow c of k space f left parenthesis x right parenthesis equals k limit as x rightwards arrow c of f left parenthesis x right parenthesis 

Dengan aturan di atas, diperoleh

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 third straight pi of fraction numerator 1 minus 2 cos space x over denominator straight pi minus 3 straight x end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 1 minus 2 cos space x over denominator left parenthesis straight pi minus 3 straight x right parenthesis end fraction times fraction numerator 1 plus 2 cos space x over denominator 1 plus 2 cos space x end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator left parenthesis 1 minus 2 cos space x right parenthesis open parentheses 1 plus 2 cos space x close parentheses over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 1 minus 4 cos squared x over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator sin squared x plus cos squared x minus 4 cos squared x over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator sin squared x minus 3 cos squared x over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator open parentheses sin space x minus square root of 3 cos space x close parentheses open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 2 open parentheses sin space x times begin display style 1 half end style minus cos space x times begin display style 1 half end style square root of 3 close parentheses open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 2 open parentheses sin space x times begin display style cos space 60 degree end style minus cos space x times sin space 60 degree close parentheses open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 2 open parentheses sin space x times begin display style cos straight pi over 3 end style minus cos space x times sin begin display style straight pi over 3 end style close parentheses open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 2 sin open parentheses x minus straight pi over 3 close parentheses open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator left parenthesis straight pi minus 3 straight x right parenthesis open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 third straight pi of fraction numerator 2 sin open parentheses x minus straight pi over 3 close parentheses open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator negative 3 open parentheses negative begin display style straight pi over 3 end style plus straight x close parentheses open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator negative 3 end fraction times limit as x rightwards arrow 1 third straight pi of fraction numerator sin open parentheses x minus straight pi over 3 close parentheses over denominator open parentheses x minus begin display style straight pi over 3 end style close parentheses end fraction times limit as x rightwards arrow 1 third straight pi of fraction numerator open parentheses sin space x plus square root of 3 cos space x close parentheses over denominator open parentheses 1 plus 2 cos space straight x close parentheses end fraction end cell row blank equals cell negative 2 over 3 times 1 times fraction numerator open parentheses sin space begin display style straight pi over 3 end style plus square root of 3 cos space straight pi over 3 close parentheses over denominator open parentheses 1 plus 2 cos space straight pi over 3 close parentheses end fraction end cell row blank equals cell negative 2 over 3 times fraction numerator open parentheses begin display style 1 half end style square root of 3 plus begin display style 1 half end style square root of 3 close parentheses over denominator open parentheses 1 plus 2 open parentheses begin display style 1 half end style close parentheses close parentheses end fraction end cell row blank equals cell negative 2 over 3 times fraction numerator square root of 3 over denominator open parentheses 1 plus 1 close parentheses end fraction end cell row blank equals cell negative 2 over 3 times fraction numerator square root of 3 over denominator 2 end fraction end cell row blank equals cell negative 1 third square root of 3 end cell end table end style  

Dengan demikian limit as x rightwards arrow 1 third straight pi of fraction numerator 1 minus 2 cos space x over denominator straight pi minus 3 straight x end fraction equals negative 1 third square root of 3.

Jadi, jawaban yang benar adalah B.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 06 Oktober 2021

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