Pertanyaan

Nilai x → ∞ lim tan x 2 8 sin x 3 = ...

Nilai $x \to \infty lim \frac{8 sin \frac{3}{x}}{tan \frac{2}{x}} = ...$

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Ingat! Diketahui . Misalkan: Maka, Sehingga, . Jadi, jawaban yang tepat adalah D.

Ingat!

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator sin space begin display style a x end style over denominator tan space b x end fraction end cell equals cell a over b end cell row cell limit as x rightwards arrow 0 of fraction numerator k times sin space begin display style x end style over denominator tan space x end fraction end cell equals cell k times limit as x rightwards arrow 0 of fraction numerator sin space begin display style x end style over denominator tan space x end fraction equals k times 1 equals k end cell end table$

Diketahui $limit as x rightwards arrow infinity of fraction numerator 8 space sin space begin display style 3 over x end style over denominator tan space begin display style 2 over x end style end fraction$ .

$limit as x rightwards arrow infinity of fraction numerator 8 space sin space begin display style 3 over x end style over denominator tan space begin display style 2 over x end style end fraction equals limit as x rightwards arrow infinity of fraction numerator 8 space sin space begin display style 3 1 over x end style over denominator tan space begin display style 2 1 over x end style end fraction$

Misalkan:

a equals 1 over x comma space maka space x equals 1 over a x rightwards arrow infinity comma space maka space a rightwards arrow 0

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator 8 space sin space begin display style 3 1 over x end style over denominator tan space begin display style 2 1 over x end style end fraction end cell equals cell limit as a rightwards arrow 0 of fraction numerator 8 space sin space begin display style 3 a end style over denominator tan space begin display style 2 a end style end fraction end cell row blank equals cell 8 times 3 over 2 end cell row blank equals 12 end table$

Sehingga, $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow infinity of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 8 space sin space begin display style 3 1 over x end style over denominator tan space begin display style 2 1 over x end style end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 12 end table$ .

Jadi, jawaban yang tepat adalah D.

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