Pertanyaan

Hasil x → ∞ lim x 2 ( 1 − cos x 4 ) = ...

Hasil $x \to \infty lim x^{2} (1 - cos \frac{4}{x}) = ...$

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: Pertama-tama, gunakan metode substitusi untuk mencari nilai limit. Limit menghasilkan bentuk tak tentu. Maka solusi limit dapat menggunakan dalil L'Hopital sebagai berikut. Dengan demikian, hasil dari adalah . Oleh karena itu, jawaban yang benar adalah E.

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.

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 .

Oleh karena itu, jawaban yang benar adalah E.