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Pertanyaan

stack text lim end text with x rightwards arrow pi below space fraction numerator sin space open parentheses x minus pi close parentheses over denominator 2 open parentheses x minus pi close parentheses plus tan space open parentheses x minus pi close parentheses end fraction equals space....

  1. negative 1 half

  2. negative 1 fourth

  3. 1 fourth

  4. 1 third

  5. 2 over 5

A. Azriah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Ingat sifat-sifat limit fungsi trigonometri berikut!

  1. stack text lim end text with x rightwards arrow c below text k end text equals text k end text
  2. stack text lim end text with x rightwards arrow 0 below fraction numerator sin space x over denominator x end fraction equals 1
  3. stack text lim end text with x rightwards arrow 0 below fraction numerator tan space x over denominator x end fraction equals 1

Berikut langkah-langkah penyelesaiannya:
Misalkan: x minus pi equals y
Jika x rightwards arrow pi, maka y rightwards arrow pi minus pi equals 0, sehingga diperoleh: 

table attributes columnalign right center left columnspacing 0px end attributes row cell stack text lim end text with x rightwards arrow pi below fraction numerator sin space open parentheses x minus pi close parentheses over denominator 2 open parentheses x minus pi close parentheses plus tan space open parentheses x minus pi close parentheses end fraction end cell equals cell stack text lim end text with y rightwards arrow 0 below fraction numerator sin space y over denominator 2 y plus tan space y end fraction end cell row blank equals cell stack text lim end text with y rightwards arrow 0 below fraction numerator sin space y over denominator 2 y plus tan space y end fraction cross times fraction numerator begin display style 1 over y end style over denominator begin display style 1 over y end style end fraction end cell row blank equals cell stack text lim end text with y rightwards arrow 0 below fraction numerator fraction numerator sin space y over denominator y end fraction over denominator fraction numerator 2 y over denominator y end fraction plus fraction numerator tan space y over denominator y end fraction end fraction end cell row blank equals cell fraction numerator stack text lim end text with y rightwards arrow 0 below fraction numerator sin space y over denominator y end fraction over denominator stack text lim end text with y rightwards arrow 0 below 2 plus stack text lim end text with y rightwards arrow 0 below fraction numerator tan space y over denominator y end fraction end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 plus 1 end fraction end cell row blank equals cell 1 third end cell end table

Oleh karena itu, jawaban yang benar adalah D.

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