Pertanyaan

x → 0 lim x 3 tan x − sin x = ...

$x \to 0 lim \frac{tan x - sin x}{x ^{3}} = ...$

N. Puspita

Master Teacher

Jawaban terverifikasi

Pembahasan

Ingat kembali identitas trigonometri, sifat limitdan limit fungsi trigonometri berikut. Dari aturan di atas, maka diperoleh Dengan demikian, . Jadi, jawaban yang benar adalah B.

Ingat kembali identitas trigonometri, sifat limit dan limit fungsi trigonometri berikut.

$tan space x equals fraction numerator sin space x over denominator cos space x end fraction$

$limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator sin space b x end fraction equals a over b$

Dari aturan di atas, maka diperoleh

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator tan space x minus sin space x over denominator x cubed end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin space x over denominator cos space x end fraction end style minus sin space x over denominator x cubed end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin space x over denominator cos space x end fraction minus fraction numerator sin space x times cos space x over denominator cos space x end fraction end style over denominator x cubed end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator begin display style fraction numerator sin space x minus sin space x times cos space x over denominator cos space x end fraction end style over denominator x cubed end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator sin space x left parenthesis 1 minus cos space x right parenthesis over denominator cos space x end fraction times 1 over x cubed end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator sin space x left parenthesis 1 minus left parenthesis 1 minus 2 space sin squared space begin display style 1 half end style x right parenthesis right parenthesis over denominator cos space x end fraction times 1 over x cubed end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator sin space x left parenthesis 2 space sin squared space begin display style 1 half end style x right parenthesis over denominator cos space x end fraction times 1 over x cubed end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator 2 over denominator cos space x end fraction times space fraction numerator sin space x over denominator x end fraction times fraction numerator sin space begin display style 1 half end style x over denominator x end fraction times fraction numerator sin space begin display style 1 half end style x over denominator x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator 2 over denominator cos space x end fraction times space limit as x rightwards arrow 0 of fraction numerator sin space x over denominator x end fraction times limit as x rightwards arrow 0 of fraction numerator sin space begin display style 1 half end style x over denominator x end fraction times limit as x rightwards arrow 0 of fraction numerator sin space begin display style 1 half end style x over denominator x end fraction end cell row blank equals cell fraction numerator 2 over denominator cos space 0 end fraction times 1 times 1 half times 1 half end cell row blank equals cell 2 over 1 times 1 times 1 half times 1 half end cell row blank equals cell 1 half end cell row blank equals cell 0 comma 5 end cell end table end style$

Dengan demikian, $limit as x rightwards arrow 0 of fraction numerator tan space x minus sin space x over denominator x cubed end fraction equals 0 comma 5$ .

Jadi, jawaban yang benar adalah B.