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Nilai x→21​πlim​1−sin2 x(sin 21​x−cos 21​x)2​ adalah ....

Pertanyaan

Nilai limit as x rightwards arrow 1 half straight pi of fraction numerator open parentheses sin space begin display style 1 half end style x minus cos space 1 half x close parentheses squared over denominator 1 minus sin squared space x end fraction adalah .... 

  1. negative 1 

  2. negative 0 comma 5 

  3. 0 

  4. 0 comma 5 

  5. 1 

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Ingat identitas trigonometri yaitu : sin squared space x plus cos squared space x equals 1.

Ingat bahwa sin space 2 x equals 2 space sin space x space cos space x maka berlaku sin space x equals 2 space sin space 1 half x space cos space 1 half x, akibatnya diperoleh :

limit as x rightwards arrow 1 half straight pi of fraction numerator open parentheses sin space begin display style 1 half end style x minus cos space begin display style 1 half end style x close parentheses squared over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator sin squared open parentheses begin display style 1 half end style x close parentheses plus cos squared space open parentheses begin display style 1 half end style x close parentheses minus 2 space sin space begin display style 1 half end style x space cos space begin display style 1 half end style x over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator 1 minus 2 space sin space begin display style 1 half end style x space cos space begin display style 1 half end style x over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator 1 minus sin space x over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator up diagonal strike 1 minus sin space x end strike over denominator up diagonal strike open parentheses 1 minus sin space x close parentheses end strike open parentheses 1 plus sin space x close parentheses end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator 1 over denominator open parentheses 1 plus sin space x close parentheses end fraction equals fraction numerator 1 over denominator 1 plus sin space open parentheses begin display style 1 half end style straight pi close parentheses end fraction equals fraction numerator 1 over denominator 1 plus 1 end fraction equals 1 half equals 0 comma 5 

Dengan demikian, nilai dari begin mathsize 12px style limit as x rightwards arrow 1 half straight pi of fraction numerator open parentheses sin space begin display style 1 half end style x minus cos space 1 half x close parentheses squared over denominator 1 minus sin squared space x end fraction equals 0 comma 5 end style.

Oleh karena itu, jawaban yang tepat adalah D.

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