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Pertanyaan

$begin mathsize 14px style lim subscript straight x rightwards arrow 3 end subscript fraction numerator 1 minus open parentheses straight x minus 2 close parentheses to the power of 8 over denominator square root of straight x plus 1 end root minus 2 end fraction equals end style$ ....

S. Difhayanti

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Perhatikan bahwa jika dilakukan substitusi , maka didapatkan hasil .

Oleh karena itu perlu dicari cara lain untuk mencari nilai limitnya.

Perhatikan perhitungan berikut!

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as straight x rightwards arrow 3 of fraction numerator 1 minus open parentheses straight x minus 2 close parentheses to the power of 8 over denominator square root of straight x plus 1 end root minus 2 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 minus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses over denominator square root of straight x plus 1 end root minus 2 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses 1 minus open parentheses straight x minus 2 close parentheses squared close parentheses over denominator square root of straight x plus 1 end root minus 2 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses close parentheses open parentheses 1 minus open parentheses straight x minus 2 close parentheses close parentheses over denominator square root of straight x plus 1 end root minus 2 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses straight x minus 1 close parentheses open parentheses negative straight x plus 3 close parentheses over denominator square root of straight x plus 1 end root minus 2 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of open parentheses fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses straight x minus 1 close parentheses open parentheses negative straight x plus 3 close parentheses over denominator square root of straight x plus 1 end root minus 2 end fraction times fraction numerator square root of straight x plus 1 end root plus 2 over denominator square root of straight x plus 1 end root plus 2 end fraction close parentheses end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses straight x minus 1 close parentheses open parentheses negative straight x plus 3 close parentheses open parentheses square root of straight x plus 1 end root plus 2 close parentheses over denominator open parentheses straight x plus 1 close parentheses minus 4 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses straight x minus 1 close parentheses open parentheses negative straight x plus 3 close parentheses open parentheses square root of straight x plus 1 end root plus 2 close parentheses over denominator straight x minus 3 end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses straight x minus 1 close parentheses open parentheses negative straight x plus 3 close parentheses open parentheses square root of straight x plus 1 end root plus 2 close parentheses over denominator negative open parentheses negative straight x plus 3 close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow 3 of fraction numerator open parentheses 1 plus open parentheses straight x minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses straight x minus 2 close parentheses squared close parentheses open parentheses straight x minus 1 close parentheses open parentheses square root of straight x plus 1 end root plus 2 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus open parentheses 3 minus 2 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses 3 minus 2 close parentheses squared close parentheses open parentheses 3 minus 1 close parentheses open parentheses square root of 3 plus 1 end root plus 2 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus open parentheses 1 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses 1 close parentheses squared close parentheses open parentheses 2 close parentheses open parentheses square root of 4 plus 2 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus 1 close parentheses open parentheses 1 plus 1 close parentheses open parentheses 2 close parentheses open parentheses 2 plus 2 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator open parentheses 2 close parentheses open parentheses 2 close parentheses open parentheses 2 close parentheses open parentheses 4 close parentheses over denominator negative 1 end fraction end cell row blank equals cell negative 32 end cell end table end style$

Jadi, jawaban yang tepat adalah E.