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Pertanyaan

limit as x rightwards arrow 0 of fraction numerator tan squared begin display style 1 over 6 end style x over denominator sin squared begin display style 1 half x end style end fraction equals....

  1. 1 half

  2. 1 over 9

  3. 1 over 6

  4. 1 third

  5. negative 1 half

A. Acfreelance

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Jawaban yang benar dari pertanyaan tersebut adalah C.

Perhatikan konsep : tan squared x equals open parentheses tan space x close parentheses squared dan sin squared x equals open parentheses sin space x close parentheses squared. Maka soal dapat diselesaikan seperti berikut :

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator tan squared begin display style 1 over 6 end style x over denominator sin squared begin display style 1 half x end style end fraction end cell equals cell limit as x rightwards arrow 0 of space open parentheses tan 1 over 6 x close parentheses squared over open parentheses sin 1 half x close parentheses squared leftwards arrow Ingat space tan space x equals fraction numerator sin space x over denominator cos space x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator open parentheses fraction numerator sin 1 over 6 x over denominator cos 1 over 6 x end fraction close parentheses over denominator open parentheses sin 1 half x close parentheses squared end fraction squared end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator begin display style open parentheses sin space 1 over 6 x close parentheses squared over open parentheses cos space 1 over 6 x close parentheses squared end style over denominator open parentheses sin 1 half x close parentheses squared end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space open parentheses sin space begin display style 1 over 6 end style x close parentheses squared over open parentheses cos space 1 over 6 x close parentheses squared times 1 over open parentheses sin begin display style 1 half end style x close parentheses squared end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator sin space begin display style 1 over 6 end style x times sin space 1 over 6 x over denominator cos space 1 over 6 x times cos space 1 over 6 x times sin space 1 half x times sin space 1 half x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator sin space 1 over 6 x over denominator sin space 1 half x end fraction times fraction numerator sin space 1 over 6 x over denominator sin space 1 half x end fraction times fraction numerator 1 over denominator cos space 1 over 6 x times cos space 1 over 6 x end fraction end cell end table

Ingat sifat limit as x rightwards arrow 0 of space f left parenthesis x right parenthesis times g left parenthesis x right parenthesis equals limit as x rightwards arrow 0 of space f left parenthesis x right parenthesis times limit as x rightwards arrow 0 of space g left parenthesis x right parenthesis dan 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, sehingga diperoleh : 

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator tan squared begin display style 1 over 6 end style x over denominator sin squared begin display style 1 half x end style end fraction end cell equals cell limit as x rightwards arrow 0 of space fraction numerator sin space 1 over 6 x over denominator sin space 1 half x end fraction times fraction numerator sin space 1 over 6 x over denominator sin space 1 half x end fraction times fraction numerator 1 over denominator cos space 1 over 6 x times cos space 1 over 6 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator sin space 1 over 6 x over denominator sin space 1 half x end fraction times limit as x rightwards arrow 0 of space fraction numerator sin space 1 over 6 x over denominator sin space 1 half x end fraction times stack lim space with x rightwards arrow 0 below fraction numerator 1 over denominator cos space 1 over 6 x times cos space 1 over 6 x end fraction end cell row blank equals cell fraction numerator begin display style 1 over 6 end style over denominator begin display style 1 half end style end fraction times fraction numerator begin display style 1 over 6 end style over denominator begin display style 1 half end style end fraction times fraction numerator 1 over denominator cos space 0 times cos space 0 end fraction end cell row blank equals cell 1 third times 1 third times fraction numerator 1 over denominator 1 times 1 end fraction end cell row blank equals cell 1 over 9 end cell end table end style

Oleh karena itu, jawaban yang benar adalah C.

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