Pertanyaan

x → 1 lim ( x − 1 ) tan ( 3 x − 3 ) cos ( 2 x − 2 ) − 1 = ...

$x \to 1 lim \frac{cos ( 2 x - 2 ) - 1}{( x - 1 ) tan ( 3 x - 3 )} = ...$

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Jawaban terverifikasi

Pembahasan

Ingat kembalibeberapa sifatberikut. Dari aturan di atas, maka diperoleh Dengan demikian, . Jadi, jawaban yang benar adalah A.

Ingat kembali beberapa sifat berikut.

$limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus 1 right parenthesis over denominator b left parenthesis x minus 1 right parenthesis end fraction equals limit as x rightwards arrow 0 of fraction numerator space a left parenthesis x minus 1 right parenthesis over denominator sin space b left parenthesis x minus 1 right parenthesis end fraction equals a over b$

$limit as x rightwards arrow c of fraction numerator sin space a left parenthesis x minus 1 right parenthesis over denominator tan space b left parenthesis x minus 1 right parenthesis end fraction equals a over b$

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 1 of fraction numerator cos space left parenthesis 2 x minus 2 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space left parenthesis 3 x minus 3 right parenthesis end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator cos space 2 left parenthesis x minus 1 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 1 minus 2 space sin squared space left parenthesis x minus 1 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative 2 space sin squared space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative 2 space sin space left parenthesis x minus 1 right parenthesis space sin space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis space tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell negative 2 limit as x rightwards arrow 1 of fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis end fraction times fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell negative 2 times limit as x rightwards arrow 1 of fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator left parenthesis x minus 1 right parenthesis end fraction limit as x rightwards arrow 1 of fraction numerator sin space left parenthesis x minus 1 right parenthesis over denominator tan space 3 left parenthesis x minus 1 right parenthesis end fraction end cell row blank equals cell negative 2 times 1 times 1 third end cell row blank equals cell negative 2 over 3 end cell end table end style$

Dengan demikian, $limit as x rightwards arrow 1 of fraction numerator cos space left parenthesis 2 x minus 2 right parenthesis minus 1 over denominator left parenthesis x minus 1 right parenthesis space tan space left parenthesis 3 x minus 3 right parenthesis end fraction equals negative 2 over 3$ .

Jadi, jawaban yang benar adalah A.