Pertanyaan

x → 0 lim x 2 1 − cos x = ...

$x \to 0 lim \frac{1 - cos x}{x ^{2}} = ...$

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

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

Ingat kembali identitas trigonometri dan limit fungsi trigonometri berikut.

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

Dari aturan di atas, maka diperoleh

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

Dengan demikian, $limit as x rightwards arrow 0 of fraction numerator 1 minus cos space x over denominator x squared end fraction equals 0 comma 5$ .

Jadi, jawaban yang benar adalah D.