Pertanyaan

x → 4 1 π lim x − 4 1 π tan x − cot x = ...

$x \to \frac{1}{4} π lim \frac{tan x - cot x}{x - \frac{1}{4} π} = ...$

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Jawaban terverifikasi

Pembahasan

Ingat kembali beberapa sifat berikut. Dari aturan di atas, maka diperoleh Misalkan . Jika maka Sehingga diperoleh Dengan demikian, . Jadi, jawaban yang benar adalah B.

Ingat kembali beberapa sifat berikut.

$tan space x equals fraction numerator sin space x over denominator cos space x end fraction space dan space c o t space x equals fraction numerator cos space x over denominator sin space x end fraction$
$limit as x rightwards arrow 0 of fraction numerator tan space a x over denominator b x end fraction equals limit as x rightwards arrow 0 of fraction numerator space a x over denominator tan space b x end fraction equals a over b$

Dari aturan di atas, maka diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator tan space x minus cot space x over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator sin space x over denominator cos space x end fraction end style minus begin display style fraction numerator cos space x over denominator sin space x end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator sin squared space x minus cos squared space x over denominator sin space x space cos space x end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator negative cos space 2 x over denominator begin display style 1 half end style times 2 times sin space x space cos space x end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style fraction numerator negative cos space 2 x over denominator begin display style 1 half times sin space 2 x end style end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style negative 2 times fraction numerator begin display style cos space 2 x end style over denominator begin display style sin space 2 x end style end fraction end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell row blank equals cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator space begin display style negative 2 times cot space 2 x end style over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell end table$

Misalkan y equals x minus 1 fourth straight pi .

Jika x rightwards arrow 1 fourth pi maka

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

Sehingga diperoleh

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 fourth straight pi of fraction numerator negative 2 space cot space 2 x over denominator x minus begin display style 1 fourth end style straight pi end fraction end cell equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space cot space 2 left parenthesis y plus begin display style 1 fourth end style straight pi right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space cot space left parenthesis 2 y plus begin display style 1 half end style straight pi right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space cot space left parenthesis begin display style 1 half end style straight pi plus 2 straight y right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator negative 2 space left parenthesis negative tan space left parenthesis 2 straight y right parenthesis right parenthesis over denominator straight y end fraction end cell row blank equals cell limit as y rightwards arrow 0 of fraction numerator 2 space tan space left parenthesis 2 straight y right parenthesis over denominator straight y end fraction end cell row blank equals cell 2 space limit as y rightwards arrow 0 of space fraction numerator tan space 2 y over denominator y end fraction end cell row blank equals cell 2 times 2 end cell row blank equals 4 end table$

Dengan demikian, $limit as x rightwards arrow 1 fourth straight pi of fraction numerator tan space x minus cot space x over denominator x minus begin display style 1 fourth end style straight pi end fraction equals 4$ .

Jadi, jawaban yang benar adalah B.