x→0lim3x3tanxsin4x=....

Pertanyaan

$limit as x rightwards arrow 0 of fraction numerator sin to the power of 4 x over denominator 3 x cubed tan x end fraction equals....$

$3$
$\frac43$
$\frac13$
$-\frac13$
$-\frac43$

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah C

Ingat sifat limit berikut $limit as x rightwards arrow 0 of fraction numerator sin x over denominator x end fraction equals 1$ .

Dengan menggunakan metode substitusi,diperoleh perhitungan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator sin to the power of 4 x over denominator 3 x cubed tan x end fraction end cell equals cell fraction numerator left parenthesis sin left parenthesis 0 right parenthesis right parenthesis to the power of 4 over denominator 3 left parenthesis 0 right parenthesis cubed. tan open parentheses 0 close parentheses end fraction end cell row blank equals cell 0 over 0 text (bentuk tak tentu) end text end cell end table$

Karena dengan metode substitusi diperoleh perhitungan dalam bentuk tak tentu ( $\frac00$ ), maka selanjutnya diselesaikan dengan cara lain yaitu dengan mengubah bentuk fungsi trigonometri sebagi berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator sin to the power of 4 x over denominator 3 x cubed tan x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator sin to the power of 4 x over denominator 3 x cubed. begin display style fraction numerator sin x over denominator cos x end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator sin to the power of 4 x over denominator begin display style fraction numerator 3 x cubed. sin x over denominator cos x end fraction end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of sin to the power of 4 x divided by fraction numerator 3 x cubed. sin x over denominator cos x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator sin to the power of 4 x. cos x over denominator 3 x cubed. sin x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator sin to the power of up diagonal strike 4 end exponent x. cos x over denominator 3 x cubed. up diagonal strike sin x end strike end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator sin cubed x. cos x over denominator 3 x cubed end fraction end cell row blank equals cell limit as x rightwards arrow 0 of open parentheses 1 third. fraction numerator sin x over denominator x end fraction. fraction numerator sin x over denominator x end fraction. fraction numerator sin x over denominator x end fraction close parentheses. limit as x rightwards arrow 0 of cos x end cell row blank blank cell open parentheses 1 third.1.1.1 close parentheses. cos open parentheses limit as x rightwards arrow 0 of x close parentheses end cell row blank equals cell 1 third. cos open parentheses 0 close parentheses end cell row blank equals cell 1 third.1 end cell row blank equals cell 1 third end cell end table$

Oleh karena itu, jawaban yang benar adalah C.