Pertanyaan

x → 1 lim ( ( 2 x − 2 ) cot ( π x − π ) ) = ...

$x \to 1 lim ((2 x - 2) cot (π x - π)) = ...$

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Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.

Pembahasan

Ingat kembali pemfaktoran dan limit fungsi trigonometri berikut. Dari aturan di atas, maka diperoleh Misalkan . Jika maka Sehingga diperoleh Dengan demikian, . Jadi, jawaban yang benar adalah E.

Ingat kembali pemfaktoran dan limit fungsi trigonometri berikut.

$fraction numerator 1 over denominator tan space x end fraction equals c o t space x$

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

Dari aturan di atas, maka diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of open parentheses open parentheses 2 straight x minus 2 close parentheses space cot space open parentheses πx minus straight pi close parentheses close parentheses end cell equals cell limit as x rightwards arrow 1 of open parentheses 2 open parentheses x minus 1 close parentheses space cot space straight pi open parentheses x minus 1 close parentheses close parentheses end cell row blank equals cell limit as x rightwards arrow 1 of open parentheses 2 open parentheses x minus 1 close parentheses fraction numerator 1 over denominator tan space straight pi open parentheses x minus 1 close parentheses space end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 2 open parentheses x minus 1 close parentheses over denominator tan space straight pi open parentheses x minus 1 close parentheses end fraction end cell end table$

Misalkan y equals x minus 1 .

Jika x rightwards arrow 1 maka

table attributes columnalign right center left columnspacing 0px end attributes row y rightwards arrow cell 1 minus 1 end cell row straight y rightwards arrow 0 end table

Sehingga diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of fraction numerator 2 left parenthesis x minus 1 right parenthesis over denominator tan space straight pi left parenthesis straight x minus 1 right parenthesis end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator 2 y over denominator tan space πy end fraction end cell row blank equals cell 2 over straight pi end cell end table$

Dengan demikian, limit as x rightwards arrow 1 of open parentheses open parentheses 2 straight x minus 2 close parentheses space cot space open parentheses πx minus straight pi close parentheses close parentheses equals 2 over straight pi .

Jadi, jawaban yang benar adalah E.