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

Pertanyaan

Nilai limit as x rightwards arrow 0 of fraction numerator 3 x times tan space 2 x times cos space x over denominator sin space 4 x times tan space x end fraction equals....

  1. 6

  2. 4

  3. 2

  4. 3 over 2

  5. 3 over 4

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah D.

Soal limit di atas merupakan limit fungsi trigonometri yang memuat sinus, cosinus dan tangen. Jika diselesaikam dengan menggunakan metode substitusi, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator 3 x times tan space 2 x times cos space x over denominator sin space 4 x times tan space x end fraction end cell equals cell fraction numerator 3 open parentheses 0 close parentheses times tan space 2 open parentheses 0 close parentheses times cos space 0 over denominator sin space 4 open parentheses 0 close parentheses times tan space 0 end fraction end cell row blank equals cell fraction numerator 0 times 0 times 1 over denominator 0 times 0 end fraction end cell row blank equals cell 0 over 0 end cell end table

Karena 0 over 0 tidak tentu, maka soal tersebut perlu diselesaikan dengan menggunakan cara lain.

Ingat bahwa pada limit fungsi trigonometri untuk tangen berlaku sifat berikut:

limit as x rightwards arrow 0 of fraction numerator x over denominator tan space x end fraction equals 1

Rumus sinus sudut rangkap, yaitu:

sin space 2 alpha equals 2 times sin space alpha times cos alpha

Kemudian terdapat identitas trigonometri:

tan space alpha equals fraction numerator sin space alpha over denominator cos space alpha end fraction

Diperoleh,

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator 3 x times tan space 2 x times cos space x over denominator sin space 4 x times tan space x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator 3 x times begin display style fraction numerator sin space 2 x over denominator cos space 2 x end fraction end style times cos space x over denominator sin space 2 open parentheses 2 x close parentheses times tan space x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 x times begin display style fraction numerator up diagonal strike sin space 2 x end strike over denominator cos space 2 x end fraction end style times cos space x over denominator 2 times up diagonal strike sin space 2 x end strike times cos space 2 x times tan space x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 x times begin display style fraction numerator 1 over denominator cos space 2 x end fraction end style times cos space x over denominator 2 times cos space 2 x times tan space x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 x over denominator tan space x end fraction times fraction numerator cos space x over denominator 2 cos squared space 2 x end fraction end cell row blank equals cell 3 over 2 open parentheses limit as x rightwards arrow 0 of fraction numerator x over denominator tan space x end fraction close parentheses open parentheses limit as x rightwards arrow 0 of fraction numerator cos space x over denominator cos squared space 2 x end fraction close parentheses end cell row blank equals cell 3 over 2 open parentheses 1 close parentheses open parentheses fraction numerator cos space 0 over denominator cos squared space 2 open parentheses 0 close parentheses end fraction close parentheses end cell row blank equals cell 3 over 2 open parentheses 1 over 1 squared close parentheses end cell row blank equals cell 3 over 2 end cell end table

Dengan demikian, nilai limit as x rightwards arrow 0 of fraction numerator 3 x times tan space 2 x times cos space x over denominator sin space 4 x times tan space x end fraction equals 3 over 2.

Oleh karena itu, jawaban yang benar adalah D.

 

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