Pertanyaan

lim x → − 1 x 2 − 1 2 − x 2 − 4 2 − x 4 = ....

$lim_{x \to - 1} \frac{2 - x ^{2} - 4 2 - x ^{4}}{x ^{2} - 1} =$ ....

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Perhatikan bahwa jika dilakukan substitusi x = -1 , maka didapatkan hasil . Sehingga perlu dicari cara lain untuk mencari nilai limitnya. Perhatikan bahwa Jadi, jawaban yang tepat adalah C.

Perhatikan bahwa jika dilakukan substitusi x = -1, maka didapatkan hasil .
Sehingga perlu dicari cara lain untuk mencari nilai limitnya.

Perhatikan bahwa

$begin mathsize 14px style limit as straight x rightwards arrow negative 1 of fraction numerator square root of 2 minus straight x squared end root minus fourth root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction equals limit as straight x rightwards arrow negative 1 of open parentheses fraction numerator square root of 2 minus straight x squared end root minus fourth root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times fraction numerator square root of 2 minus straight x squared end root plus fourth root of 2 minus straight x to the power of 4 end root over denominator square root of 2 minus straight x squared end root plus fourth root of 2 minus straight x to the power of 4 end root end fraction close parentheses equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator open parentheses straight x squared minus 1 close parentheses open parentheses square root of 2 minus straight x squared end root plus fourth root of 2 minus straight x to the power of 4 end root close parentheses end fraction equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times limit as straight x rightwards arrow negative 1 of fraction numerator 1 over denominator square root of 2 minus straight x squared end root plus fourth root of 2 minus straight x to the power of 4 end root end fraction equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator square root of 2 minus open parentheses negative 1 close parentheses squared end root plus fourth root of 2 minus open parentheses negative 1 close parentheses to the power of 4 end root end fraction equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator square root of 2 minus 1 end root plus fourth root of 2 minus 1 end root end fraction equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator square root of 1 plus fourth root of 1 end fraction equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator 1 plus 1 end fraction equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times 1 half equals limit as straight x rightwards arrow negative 1 of open parentheses fraction numerator open parentheses 2 minus straight x squared close parentheses minus square root of 2 minus straight x to the power of 4 end root over denominator straight x squared minus 1 end fraction times fraction numerator open parentheses 2 minus straight x squared close parentheses plus square root of 2 minus straight x to the power of 4 end root over denominator open parentheses 2 minus straight x squared close parentheses plus square root of 2 minus straight x to the power of 4 end root end fraction close parentheses times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses squared minus open parentheses 2 minus straight x to the power of 4 close parentheses over denominator open parentheses straight x squared minus 1 close parentheses open parentheses open parentheses 2 minus straight x squared close parentheses plus square root of 2 minus straight x to the power of 4 end root close parentheses end fraction times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 2 minus straight x squared close parentheses squared minus open parentheses 2 minus straight x to the power of 4 close parentheses over denominator straight x squared minus 1 end fraction times limit as straight x rightwards arrow negative 1 of fraction numerator 1 over denominator open parentheses 2 minus straight x squared close parentheses plus square root of 2 minus straight x to the power of 4 end root end fraction times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator open parentheses 4 minus 4 straight x squared plus straight x to the power of 4 close parentheses minus 2 plus straight x to the power of 4 over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator open parentheses 2 minus open parentheses negative 1 close parentheses squared close parentheses plus square root of 2 minus open parentheses negative 1 close parentheses to the power of 4 end root end fraction times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator 2 straight x to the power of 4 minus 4 straight x squared plus 2 over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator open parentheses 2 minus 1 close parentheses plus square root of 2 minus 1 end root end fraction times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator 2 open parentheses straight x to the power of 4 minus 2 straight x squared plus 1 close parentheses over denominator straight x squared minus 1 end fraction times fraction numerator 1 over denominator 1 plus square root of 1 end fraction times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator 2 open parentheses straight x squared minus 1 close parentheses squared over denominator straight x squared minus 1 end fraction times 1 half times 1 half equals limit as straight x rightwards arrow negative 1 of fraction numerator 2 open parentheses straight x squared minus 1 close parentheses squared over denominator straight x squared minus 1 end fraction times 1 half times 1 half equals limit as straight x rightwards arrow negative 1 of 2 open parentheses straight x squared minus 1 close parentheses times 1 half times 1 half equals 2 open parentheses open parentheses negative 1 close parentheses squared minus 1 close parentheses times 1 half times 1 half equals 2 open parentheses 1 minus 1 close parentheses times 1 half times 1 half equals 2 open parentheses 0 close parentheses times 1 half times 1 half equals 0 end style$

Jadi, jawaban yang tepat adalah C.