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Hasil dari x→0lim​1−cos2 3xx2​=....

Pertanyaan

Hasil dari limit as x rightwards arrow 0 of fraction numerator x squared over denominator 1 minus cos squared space 3 x end fraction equals....

  1. 1 over 9

  2. 1 third

  3. 0

  4. negative 1 over 9

  5. negative 1 third

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah A.

Soal limit tersebut merupakan limit fungsi trigonometri yang memuat cosinus. Jika diselesaikan 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 x squared over denominator 1 minus cos squared space 3 x end fraction end cell equals cell fraction numerator open parentheses 0 close parentheses squared over denominator 1 minus cos squared subscript blank space 3 open parentheses 0 close parentheses end fraction end cell row blank equals cell fraction numerator 0 over denominator 1 minus cos squared space 0 end fraction end cell row blank equals cell fraction numerator 0 over denominator 1 minus 1 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 terdapat identitas trigonometri:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin squared space alpha plus cos squared space alpha end cell equals 1 row cell sin squared space alpha end cell equals cell 1 minus cos squared space alpha end cell end table

Pada limit fungsi trigonometri yang memuat sinus berlaku:

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

Diperoleh,

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator x squared over denominator 1 minus cos squared space 3 x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator x squared over denominator sin squared space 3 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator x over denominator sin space 3 x end fraction times fraction numerator x over denominator sin space 3 x end fraction end cell row blank equals cell 1 third limit as x rightwards arrow 0 of fraction numerator 3 x over denominator sin space 3 x end fraction times 1 third limit as x rightwards arrow 0 of fraction numerator 3 x over denominator sin space 3 x end fraction end cell row blank equals cell 1 third open parentheses 1 close parentheses times 1 third open parentheses 1 close parentheses end cell row blank equals cell 1 over 9 end cell end table

Dengan demikian, hasil dari limit as x rightwards arrow 0 of fraction numerator x squared over denominator 1 minus cos squared space 3 x end fraction equals 1 over 9.

Oleh karena itu, jawaban yang benar adalah A.

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