Pertanyaan

Nilai dari adalah ....

Nilai dari $begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator cot squared invisible function application open parentheses fraction numerator 3 over denominator square root of x end fraction close parentheses over denominator 6 x end fraction end style$ adalah ....

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Misal . Akibatnya, ketika ,maka . Karena , didapat . Oleh karena itu, didapat perhitungan sebagai berikut. Dapat diperhatikan bahwa ketika , maka . Oleh karena itu, dapat kita tuliskan menjadi . Kemudian, ingat bahwa . Akibatnya, didapat nilai adalah sebagai berikut. Jadi, jawaban yang tepat adalah E.

Misal $begin mathsize 14px style y equals space fraction numerator 1 over denominator square root of x end fraction end style$ . Akibatnya, ketika , maka .

Karena $begin mathsize 14px style y equals space fraction numerator 1 over denominator square root of x end fraction end style$ , didapat .

Oleh karena itu, didapat perhitungan sebagai berikut.

$begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator cot squared invisible function application open parentheses fraction numerator 3 over denominator square root of x end fraction close parentheses over denominator 6 x end fraction equals limit as x rightwards arrow straight infinity of invisible function application open parentheses fraction numerator 1 over denominator 6 x end fraction times cot squared invisible function application 3 open parentheses fraction numerator 1 over denominator square root of x end fraction close parentheses close parentheses equals limit as x rightwards arrow straight infinity of invisible function application open parentheses 1 over 6 times 1 over x times cot squared invisible function application 3 open parentheses fraction numerator 1 over denominator square root of x end fraction close parentheses close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times fraction numerator 1 over denominator 1 over y squared end fraction times cot squared invisible function application 3 y close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times y squared times cot squared invisible function application 3 y close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times y squared times fraction numerator 1 over denominator tan squared invisible function application 3 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times fraction numerator y squared over denominator tan squared invisible function application 3 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times fraction numerator y times y over denominator tan invisible function application 3 y times tan invisible function application 3 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times fraction numerator y over denominator tan invisible function application 3 y end fraction times fraction numerator y over denominator tan invisible function application 3 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times fraction numerator y over denominator tan invisible function application 3 y end fraction times 3 over 3 times fraction numerator y over denominator tan invisible function application 3 y end fraction times 3 over 3 close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 6 times fraction numerator 3 y over denominator tan invisible function application 3 y end fraction times 1 third times fraction numerator 3 y over denominator tan invisible function application 3 y end fraction times 1 third close parentheses equals limit as y rightwards arrow 0 of invisible function application open parentheses 1 over 54 times fraction numerator 3 y over denominator tan invisible function application 3 y end fraction times fraction numerator 3 y over denominator tan invisible function application 3 y end fraction close parentheses equals limit as y rightwards arrow 0 of invisible function application 1 over 54 times limit as y rightwards arrow 0 of fraction numerator 3 y over denominator tan invisible function application 3 y end fraction times limit as y rightwards arrow 0 of fraction numerator 3 y over denominator tan invisible function application 3 y end fraction equals 1 over 54 times limit as y rightwards arrow 0 of fraction numerator 3 y over denominator sin invisible function application 3 y end fraction times limit as y rightwards arrow 0 of fraction numerator 3 y over denominator sin invisible function application 3 y end fraction end style$

Dapat diperhatikan bahwa ketika , maka . Oleh karena itu, $begin mathsize 14px style limit as y rightwards arrow 0 of fraction numerator 3 y over denominator tan 3 y end fraction end style$ dapat kita tuliskan menjadi $begin mathsize 14px style limit as 3 y rightwards arrow 0 of fraction numerator 3 y over denominator tan 3 y end fraction end style$ .

Kemudian, ingat bahwa $begin mathsize 14px style limit as theta rightwards arrow 0 of fraction numerator theta over denominator tan theta end fraction equals 1 end style$ .

Akibatnya, didapat nilai $begin mathsize 14px style limit as x rightwards arrow straight infinity of invisible function application fraction numerator cot squared invisible function application open parentheses fraction numerator 3 over denominator square root of x end fraction close parentheses over denominator 6 x end fraction end style$ adalah sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow straight infinity of invisible function application fraction numerator cot squared invisible function application open parentheses fraction numerator 3 over denominator square root of x end fraction close parentheses over denominator 6 x end fraction end cell equals cell 1 over 54 times limit as 3 y rightwards arrow 0 of fraction numerator 3 y over denominator sin invisible function application 3 y end fraction times limit as 3 y rightwards arrow 0 of fraction numerator 3 y over denominator sin invisible function application 3 y end fraction end cell row blank equals cell 1 over 54 times 1 times 1 end cell row blank equals cell 1 over 54 end cell end table end style$

Jadi, jawaban yang tepat adalah E.