Nilai x→0limx21−cos 4x=...

Pertanyaan

Nilai $limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 4 x over denominator x squared end fraction equals...$

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

Sebelumnya ingat dalil L'Hopital untuk mencari limit:

$limit as x rightwards arrow a of fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction equals limit as x rightwards arrow a of fraction numerator f apostrophe open parentheses x close parentheses over denominator g apostrophe open parentheses x close parentheses end fraction$

Berdasarkan dalil L'Hopital, maka:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 4 x over denominator x squared end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d over denominator d x end fraction end style open parentheses 1 minus cos space 4 x close parentheses over denominator begin display style fraction numerator d over denominator d x end fraction end style open parentheses x squared close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator begin display style 0 minus open parentheses negative 4 space sin space 4 x close parentheses end style over denominator begin display style 2 x end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 4 space sin space 4 x over denominator 2 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 2 space sin space 4 x over denominator x end fraction end cell row blank equals cell 2 limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator d over denominator d x end fraction end style open parentheses sin space 4 x close parentheses over denominator begin display style fraction numerator d over denominator d x end fraction end style open parentheses x close parentheses end fraction end cell row blank equals cell 2 times limit as x rightwards arrow 0 of fraction numerator 4 space cos space 4 x over denominator 1 end fraction end cell row blank equals cell 2 times fraction numerator 4 times cos space 4 open parentheses 0 close parentheses over denominator 1 end fraction end cell row blank equals cell 2 times fraction numerator 4 times 1 over denominator 1 end fraction end cell row blank equals 8 end table$

Dengan demikian, hasil dari $limit as x rightwards arrow 0 of fraction numerator 1 minus cos space 4 x over denominator x squared end fraction$ adalah .

Oleh karena itu, jawaban yang benar adalah B.