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limx→−1​x2−12−x2​−42−x4​​= ....

Pertanyaan

begin mathsize 14px style lim subscript straight x rightwards arrow negative 1 end subscript 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 end style ....

  1. size 14px 1 

  2. begin mathsize 14px style 1 half end style 

  3. size 14px 0 

  4. undefined 

  5. begin mathsize 14px style negative 1 end style 

Pembahasan:

Perhatikan bahwa jika dilakukan substitusi = -1, maka didapatkan hasil undefined .
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.

Jawaban terverifikasi

Dijawab oleh:

Terakhir diupdate 04 Oktober 2021

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