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Pertanyaan

limit as x rightwards arrow 0 of fraction numerator square root of 1 plus sin space x end root minus square root of 1 minus sin space x end root over denominator x end fraction equals...      

  1. 1   

  2. 0 comma 75 

  3. 0 comma 5   

  4. 0 comma 25 

  5. 0   

Pembahasan Video:

Pembahasan Soal:

Ingat kembali faktorisasi aljabar, sifat limit dan limit fungsi trigonometri berikut.

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

 

  • 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 0 of fraction numerator sin space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator sin space b x end fraction equals a over b   

Dari aturan di atas, maka diperoleh

==========limx0x1+sinx1sinxlimx0(x1+sinx1sinx)(1+sinx+1sinx1+sinx+1sinx)limx0x(1+sinx+1sinx)(1+sinx)2(1sinx)2limx0x(1+sinx+1sinx)(1+sinx)(1sinx)limx0x(1+sinx+1sinx)2sinxlimx0xsinxlimx01+sinx+1sinx211+sin0+1sin0211+0+10211+121221             

Dengan demikian, limit as x rightwards arrow 0 of fraction numerator square root of 1 plus sin space x end root minus square root of 1 minus sin space x end root over denominator x end fraction equals 1.

Jadi, jawaban yang benar adalah A.

 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 13 September 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pembahasan Soal:

Ingat kembali beberapa sifat berikut.

  • 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 0 of fraction numerator space 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

 

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

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 x end cell equals cell 1 minus 2 space sin squared space x end cell row cell cos space x end cell equals cell 1 minus 2 space sin squared space 1 half x end cell end table 

​​​​​​​

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

 

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

Dengan aturan di atas diperoleh.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction times fraction numerator 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus cos left parenthesis x minus 2 right parenthesis over denominator left parenthesis x squared minus 4 x plus 4 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus open parentheses 1 minus 2 sin squared begin display style 1 half end style left parenthesis x minus 2 right parenthesis close parentheses over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 1 minus 1 plus 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin squared 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator 2 sin 1 half left parenthesis x minus 2 right parenthesis times sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis left parenthesis begin display style x minus 2 end style right parenthesis left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell 2 times limit as x rightwards arrow 2 of fraction numerator sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis end fraction times limit as x rightwards arrow 2 of fraction numerator sin 1 half left parenthesis x minus 2 right parenthesis over denominator left parenthesis x minus 2 right parenthesis end fraction times limit as x rightwards arrow 2 of fraction numerator 1 over denominator left parenthesis 1 plus square root of cos left parenthesis x minus 2 right parenthesis end root right parenthesis end fraction end cell row blank equals cell 2 times 1 half times 1 half times fraction numerator 1 over denominator 1 plus square root of cos left parenthesis 2 minus 2 right parenthesis end root end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus square root of cos left parenthesis 0 right parenthesis end root end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus square root of 1 end fraction end cell row blank equals cell 1 half times fraction numerator 1 over denominator 1 plus 1 end fraction end cell row blank equals cell 1 half times 1 half end cell row blank equals cell 1 fourth end cell end table  

Jadi, limit as x rightwards arrow 2 of fraction numerator 1 minus square root of cos left parenthesis x minus 2 right parenthesis end root over denominator x squared minus 4 x plus 4 end fraction equals 1 fourth.

0

Roboguru

Pembahasan Soal:

Ingat kembali selisih cosinus, sifat limit dan limit fungsi trigonometri berikut.

  • cos space A minus cos space B equals negative 2 space sin space left parenthesis fraction numerator A plus B over denominator 2 end fraction right parenthesis space sin space left parenthesis fraction numerator A minus B over denominator 2 end fraction right parenthesis 

 

  • 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

 

  • limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator sin space b x end fraction equals a over b

​​​​​​​

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

Dari aturan di atas, maka diperoleh

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator cos space m x minus cos space n x over denominator x squared end fraction end cell equals cell limit as x rightwards arrow 0 of space fraction numerator negative 2 space sin space left parenthesis begin display style fraction numerator m x plus n x over denominator 2 end fraction end style right parenthesis space sin space left parenthesis begin display style fraction numerator m x minus n x over denominator 2 end fraction end style right parenthesis over denominator x squared end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator negative 2 space sin space left parenthesis begin display style fraction numerator m plus n over denominator 2 end fraction end style right parenthesis x space times space sin space left parenthesis begin display style fraction numerator m minus n over denominator 2 end fraction end style right parenthesis x over denominator x times x end fraction end cell row blank equals cell negative 2 space limit as x rightwards arrow 0 of space fraction numerator sin space left parenthesis fraction numerator m plus n over denominator 2 end fraction right parenthesis x over denominator x end fraction times limit as x rightwards arrow 0 of space fraction numerator sin space left parenthesis fraction numerator m minus n over denominator 2 end fraction right parenthesis x over denominator x end fraction end cell row blank equals cell negative 2 times left parenthesis fraction numerator m plus n over denominator 2 end fraction right parenthesis times left parenthesis fraction numerator m minus n over denominator 2 end fraction right parenthesis end cell row blank equals cell negative 1 half left parenthesis m plus n right parenthesis left parenthesis m minus n right parenthesis end cell row blank equals cell negative 1 half left parenthesis m squared minus n squared right parenthesis end cell row blank equals cell 1 half left parenthesis negative m squared plus n squared right parenthesis end cell row blank equals cell 1 half left parenthesis n squared minus m squared right parenthesis end cell end table end style       

Dengan demikian, limit as x rightwards arrow 0 of fraction numerator cos space m x minus cos space n x over denominator x squared end fraction equals 1 half left parenthesis n squared minus m squared right parenthesis.

Jadi, jawaban yang benar adalah A.

 

0

Roboguru

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.

0

Roboguru

Pembahasan Soal:

limit as x rightwards arrow 0 of fraction numerator 2 x squared plus x over denominator sin x end fraction equals limit as x rightwards arrow 0 of fraction numerator x left parenthesis 2 x plus 1 right parenthesis over denominator sin x end fraction  space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of fraction numerator x over denominator sin x end fraction limit as x rightwards arrow 0 of 2 x plus 1  space space space space space space space space space space space space space space space space space space space space space equals 1 left parenthesis 1 right parenthesis  space space space space space space space space space space space space space space space space space space space space space equals 1

2

Roboguru

Pembahasan Soal:

limit as x rightwards arrow 2 of fraction numerator left parenthesis t squared minus 5 t plus 6 right parenthesis sin left parenthesis t minus 2 right parenthesis over denominator left parenthesis t squared minus t minus 2 right parenthesis squared end fraction  equals limit as x rightwards arrow 2 of fraction numerator left parenthesis t squared minus 5 t plus 6 right parenthesis sin left parenthesis t minus 2 right parenthesis over denominator left parenthesis left parenthesis t minus 2 right parenthesis left parenthesis t plus 1 right parenthesis right parenthesis squared end fraction  equals limit as x rightwards arrow 2 of fraction numerator left parenthesis t minus 3 right parenthesis left parenthesis t minus 2 right parenthesis sin left parenthesis t minus 2 right parenthesis over denominator left parenthesis t minus 2 right parenthesis squared left parenthesis t plus 1 right parenthesis squared end fraction  equals limit as x rightwards arrow 2 of fraction numerator left parenthesis t minus 3 right parenthesis left parenthesis t minus 2 right parenthesis over denominator left parenthesis t minus 2 right parenthesis left parenthesis t plus 1 right parenthesis squared end fraction limit as x rightwards arrow 2 of fraction numerator sin left parenthesis t minus 2 right parenthesis over denominator left parenthesis t minus 2 right parenthesis end fraction  equals fraction numerator left parenthesis 2 minus 3 right parenthesis over denominator left parenthesis 2 plus 1 right parenthesis squared end fraction left parenthesis 1 right parenthesis  equals negative 1 over 9

1

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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