Pertanyaan

x → 0 lim x 2 cos m x − cos n x = ...

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

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Jawaban terverifikasi

Pembahasan

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

Ingat kembali selisih cosinus, sifat limit dan limit fungsi trigonometri berikut.

$cos space A minus cos space B equals negative 2 space sin space left parenthesis fraction numerator A plus B over denominator 2 end fraction right parenthesis space sin space left parenthesis fraction numerator A minus B over denominator 2 end fraction right parenthesis$

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

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 cos space m x minus cos space n x over denominator x squared end fraction end cell equals cell limit as x rightwards arrow 0 of space fraction numerator negative 2 space sin space left parenthesis begin display style fraction numerator m x plus n x over denominator 2 end fraction end style right parenthesis space sin space left parenthesis begin display style fraction numerator m x minus n x over denominator 2 end fraction end style right parenthesis over denominator x squared end fraction end cell row blank equals cell limit as x rightwards arrow 0 of space fraction numerator negative 2 space sin space left parenthesis begin display style fraction numerator m plus n over denominator 2 end fraction end style right parenthesis x space times space sin space left parenthesis begin display style fraction numerator m minus n over denominator 2 end fraction end style right parenthesis x over denominator x times x end fraction end cell row blank equals cell negative 2 space limit as x rightwards arrow 0 of space fraction numerator sin space left parenthesis fraction numerator m plus n over denominator 2 end fraction right parenthesis x over denominator x end fraction times limit as x rightwards arrow 0 of space fraction numerator sin space left parenthesis fraction numerator m minus n over denominator 2 end fraction right parenthesis x over denominator x end fraction end cell row blank equals cell negative 2 times left parenthesis fraction numerator m plus n over denominator 2 end fraction right parenthesis times left parenthesis fraction numerator m minus n over denominator 2 end fraction right parenthesis end cell row blank equals cell negative 1 half left parenthesis m plus n right parenthesis left parenthesis m minus n right parenthesis end cell row blank equals cell negative 1 half left parenthesis m squared minus n squared right parenthesis end cell row blank equals cell 1 half left parenthesis negative m squared plus n squared right parenthesis end cell row blank equals cell 1 half left parenthesis n squared minus m squared right parenthesis end cell end table end style$

Dengan demikian, $limit as x rightwards arrow 0 of fraction numerator cos space m x minus cos space n x over denominator x squared end fraction equals 1 half left parenthesis n squared minus m squared right parenthesis$ .

Jadi, jawaban yang benar adalah A.

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