Hasil x→∞lim​x2(1−cos x4​)=...

Pertanyaan

Hasil limit as x rightwards arrow infinity of x squared open parentheses 1 minus cos space 4 over x close parentheses equals...

  1. 1 fourth

  2. 1 half

  3. 2

  4. 4

  5. 8

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.

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

Pertama-tama, gunakan metode substitusi untuk mencari nilai limit.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of x squared open parentheses 1 minus cos space 4 over x close parentheses end cell equals cell infinity open parentheses 1 minus cos space 4 over infinity close parentheses end cell row blank equals cell infinity open parentheses 1 minus cos space 0 close parentheses end cell row blank equals cell infinity open parentheses 1 minus 1 close parentheses end cell row blank equals cell infinity times 0 space left parenthesis bentuk space tak space tentu right parenthesis end cell end table

Limit menghasilkan bentuk tak tentu. Maka solusi limit dapat menggunakan dalil L'Hopital sebagai berikut.

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

Dengan demikian, hasil dari limit as x rightwards arrow infinity of x squared open parentheses 1 minus cos space 4 over x close parentheses  adalah 8.

Oleh karena itu, jawaban yang benar adalah E.

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