Pertanyaan

x → 3 1 π lim π − 3 x 1 − 2 cos x = …

$x \to \frac{1}{3} π lim \frac{1 - 2 cos x}{π - 3 x} = \dots$

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

Ingat kembali pemfaktoran, identitas trigonometri, selisih dua sudut pada sin dan limit fungsi trigonometri berikut ini. Dengan aturan di atas, diperoleh Dengan demikian . Jadi, jawaban yang benar adalah B.

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

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

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.