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Hasil x→0lim​cos 8x−cos 2x5x tan 3x​ adalah ...

Pertanyaan

Hasil limit as x rightwards arrow 0 of fraction numerator 5 x space tan space 3 x over denominator cos space 8 x minus cos space 2 x end fraction adalah ...

  1. 2

  2. 1

  3. 1 half

  4. negative 1 half

  5. negative 1

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Rumus limit trigonometri untuk x rightwards arrow 0 yang didapat dari penurunan dengan kaidah L'Hopital sebagai berikut.

left parenthesis straight i right parenthesis space limit as x rightwards arrow 0 of fraction numerator a x over denominator sin space b x end fraction equals a over b left parenthesis ii right parenthesis space limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator sin space b x end fraction equals a over b

Selain itu, ingatlah identitas trigonometri berikut.

left parenthesis straight i right parenthesis space tan space x equals fraction numerator sin space x over denominator cos space x end fraction left parenthesis ii right parenthesis space sin space 2 x equals 2 space sin space x space cos space x

Pertama, lakukan metode substitusi untuk menjawab soal limit di atas.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator 5 x space tan space 3 x over denominator cos space 8 x minus cos space 2 x end fraction end cell equals cell fraction numerator 5 open parentheses 0 close parentheses space tan space 3 open parentheses 0 close parentheses over denominator cos space 8 left parenthesis 0 right parenthesis minus cos space 2 left parenthesis 0 right parenthesis end fraction end cell row blank equals cell fraction numerator 0 times 0 over denominator 1 minus 1 end fraction end cell row blank equals cell 0 over 0 left parenthesis bentuk space tak space tentu right parenthesis end cell end table

Karena menghasilkan bentuk tak tentu, penyelesaian limit di atas adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 0 of fraction numerator 5 x space tan space 3 x over denominator cos space 8 x minus cos space 2 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space 5 times limit as x rightwards arrow 0 of open parentheses x times fraction numerator sin space 3 x over denominator cos space 3 x end fraction times fraction numerator space 1 over denominator cos space 8 x minus co s space 2 x end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 0 of space 5 times limit as x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator cos space 3 x end fraction close parentheses times limit as x rightwards arrow 0 of open parentheses fraction numerator begin display style fraction numerator d over denominator d x end fraction end style open parentheses x space sin space 3 x close parentheses over denominator begin display style fraction numerator d over denominator d x end fraction end style open parentheses cos space 8 x minus cos space 2 x close parentheses end fraction close parentheses end cell row blank equals cell 5 times fraction numerator 1 over denominator cos space 3 left parenthesis 0 right parenthesis end fraction times limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x plus 3 x space cos space 3 x over denominator negative 8 space sin space 8 x plus 2 space sin space 2 x end fraction close parentheses end cell row blank equals cell 5 times limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x plus 3 x space cos space 3 x over denominator negative 8 open parentheses 2 space sin space 4 x space cos space 4 x close parentheses plus 2 times 2 space sin space x space cos space x end fraction close parentheses end cell row blank equals cell 5 times limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x plus 3 x space cos space 3 x over denominator negative 16 open parentheses 2 space sin space 2 x space cos space 2 x close parentheses space cos space 4 x plus 4 space sin space x space cos space x end fraction close parentheses end cell row blank equals cell 5 times limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x plus 3 x space cos space 3 x over denominator negative 32 open parentheses 2 space sin space x space cos space x close parentheses space cos space 2 x space cos space 4 x plus 4 space sin space x space cos space x end fraction close parentheses end cell row blank equals cell 5 times limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x plus 3 x space cos space 3 x over denominator negative 64 space sin space x space cos space x space cos space 2 x space cos space 4 x plus 4 space sin space x space cos space x end fraction close parentheses end cell row blank equals cell 5 times limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x plus 3 x space cos space 3 x over denominator negative 4 space sin space x space cos space x open parentheses 16 space cos space 2 x space cos space 4 x minus 1 close parentheses end fraction close parentheses end cell row blank equals cell 5 times open square brackets limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x over denominator negative 4 space sin space x space cos space x open parentheses 16 space cos space 2 x space cos space 4 x minus 1 close parentheses end fraction close parentheses plus limit as x rightwards arrow 0 of open parentheses fraction numerator 3 x space cos space 3 x over denominator negative 4 space sin space x space cos space x open parentheses 16 space cos space 2 x space cos space 4 x minus 1 close parentheses end fraction close parentheses close square brackets end cell row blank equals cell 5 times open square brackets limit as x rightwards arrow 0 of open parentheses fraction numerator sin space 3 x over denominator negative 4 space sin space x end fraction close parentheses times limit as x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator cos space x open parentheses 16 space cos space 2 x space cos space 4 x minus 1 close parentheses end fraction close parentheses plus limit as x rightwards arrow 0 of open parentheses fraction numerator 3 x over denominator negative 4 space sin space x end fraction close parentheses times limit as x rightwards arrow 0 of open parentheses fraction numerator cos space 3 x over denominator cos space x end fraction close parentheses times limit as x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator open parentheses 16 space cos space 2 x space cos space 4 x minus 1 close parentheses end fraction close parentheses close square brackets end cell row blank equals cell 5 times open square brackets negative 3 over 4 times fraction numerator 1 over denominator 1 open parentheses 16 minus 1 close parentheses end fraction plus open parentheses negative 3 over 4 close parentheses times 1 over 1 times fraction numerator 1 over denominator 16 minus 1 end fraction close square brackets end cell row blank equals cell 5 times open square brackets negative 3 over 4 times 1 over 15 plus open parentheses negative 3 over 60 close parentheses close square brackets end cell row blank equals cell 5 times open square brackets negative 1 over 20 plus open parentheses negative 1 over 20 close parentheses close square brackets end cell row blank equals cell 5 times open square brackets negative 2 over 20 close square brackets end cell row blank equals cell 5 times open square brackets negative 1 over 10 close square brackets end cell row blank equals cell negative 5 over 10 end cell row blank equals cell negative 1 half end cell end table

Dengan demikian, hasil dari limit as x rightwards arrow 0 of fraction numerator 5 x space tan space 3 x over denominator cos space 8 x minus cos space 2 x end fraction adalah negative 1 half.

Oleh karena itu, jawaban yang benar adalah D.

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