Pertanyaan

x → 0 lim x + sin 5 x tan 2 x + sin x = ...

$x \to 0 lim \frac{tan 2 x + sin x}{x + sin 5 x} = ...$

P. Anggrayni

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

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

Ingat kembali sifat limit dan limit fungsi trigonometri berikut.

$limit as x rightwards arrow c of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals fraction numerator limit as x rightwards arrow c of space f left parenthesis x right parenthesis over denominator limit as x rightwards arrow c of space g left parenthesis x right parenthesis end fraction$
$limit as x rightwards arrow 0 of fraction numerator sin space a x over denominator b x end fraction equals a over b$
$limit as x rightwards arrow 0 of fraction numerator tan space a x over denominator b x end fraction equals a over b$

Dari aturan di atas, maka diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator tan space 2 x plus sin space x over denominator x plus sin space 5 x end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses tan space 2 x plus sin space x close parentheses over denominator open parentheses x plus sin space 5 x close parentheses end fraction times fraction numerator begin display style 1 over x end style over denominator begin display style 1 over x end style end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator begin display style fraction numerator tan space 2 x over denominator x end fraction end style plus begin display style fraction numerator sin space x over denominator x end fraction end style over denominator begin display style x over x end style plus begin display style fraction numerator sin space 5 x over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator limit as x rightwards arrow 0 of space open parentheses begin display style fraction numerator tan space 2 x over denominator x end fraction end style plus begin display style fraction numerator sin space x over denominator x end fraction end style close parentheses over denominator limit as x rightwards arrow 0 of space open parentheses begin display style x over x end style plus begin display style fraction numerator sin space x over denominator x end fraction end style close parentheses end fraction end cell row blank equals cell fraction numerator limit as x rightwards arrow 0 of begin display style fraction numerator tan space 2 x over denominator x end fraction end style plus limit as x rightwards arrow 0 of begin display style fraction numerator sin space x over denominator x end fraction end style over denominator limit as x rightwards arrow 0 of begin display style x over x end style plus limit as x rightwards arrow 0 of begin display style fraction numerator sin space x over denominator x end fraction end style end fraction end cell row blank equals cell fraction numerator 2 plus 1 over denominator 1 plus 5 end fraction end cell row blank equals cell 3 over 6 end cell row blank equals cell 1 half end cell row blank equals cell 0 comma 5 end cell end table$

Dengan demikian, $limit as x rightwards arrow 0 of fraction numerator tan space 2 x plus sin space x over denominator x plus sin space 5 x end fraction equals 0 comma 5$ .

Jadi, jawaban yang benar adalah B.