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Tentukan nilai x→0lim​sin 2x⋅tan 2x1−cos 6x​!

Pertanyaan

Tentukan nilai limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 6 x over denominator sin space 2 x times tan space 2 x end fraction!

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

nilai dari limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 6 x over denominator sin space 2 x times tan space 2 x end fraction adalah 9 over 2.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold 9 over bold 2.

Ingat!

  • cos space 2 x equals 1 minus 2 space sin squared space x
  • 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
  • limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator tan space b x end fraction equals a over b
  • sifat limit: limit as x rightwards arrow 0 of left parenthesis f left parenthesis x right parenthesis times f left parenthesis g right parenthesis equals limit as x rightwards arrow 0 of f left parenthesis x right parenthesis times limit as x rightwards arrow 0 of f left parenthesis g right parenthesis

Apabila nilai x equals 0 langsung di subsitusi ke fungsi fraction numerator 1 minus cos space 6 x over denominator sin space 2 x times tan space 2 x end fraction maka akan di peroleh nilai limit 1 over 0. Oleh karena itu harus disederhanakan terlebih dahulu. 

Berdasarkan cos space 2 x equals 1 minus 2 space sin squared space x, maka cos space 6 x equals 1 minus 2 space sin squared space 3 x, sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 6 x over denominator sin space 2 x times tan space 2 x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator 1 minus left parenthesis 1 minus 2 space sin squared space 3 x right parenthesis over denominator sin space 2 x times tan space 2 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 2 space sin squared space 3 x over denominator sin space 2 x times tan space 2 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 2 space sin space 3 x times sin space 3 x over denominator sin space 2 x times tan space 2 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of open parentheses 2 times fraction numerator sin space 3 x over denominator sin space 2 x end fraction times fraction numerator sin space 3 x over denominator tan space 2 x end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 0 of 2 times limit as x rightwards arrow 0 of fraction numerator sin space 3 x over denominator sin space 2 x end fraction times limit as x rightwards arrow 0 of fraction numerator sin space 3 x over denominator tan space 2 x end fraction end cell row blank equals cell 2 times 3 over 2 times 3 over 2 end cell row blank equals cell 9 over 2 end cell end table

Dengan demikian nilai dari limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 6 x over denominator sin space 2 x times tan space 2 x end fraction adalah 9 over 2.

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