limx→3​6x−9​−x2−2x+6​3x2−11​−3x+7​​= ....

Pertanyaan

begin mathsize 14px style lim subscript straight x rightwards arrow 3 end subscript fraction numerator square root of 3 straight x squared minus 11 end root minus square root of 3 straight x plus 7 end root over denominator square root of 6 straight x minus 9 end root minus square root of straight x squared minus 2 straight x plus 6 end root end fraction equals end style ....

  1. begin mathsize 14px style 15 over 16 end style 

  2. begin mathsize 14px style 15 over 8 end style 

  3. begin mathsize 14px style 45 over 16 end style 

  4. begin mathsize 14px style 45 over 8 end style 

  5. begin mathsize 14px style 45 over 4 end style 

R. RGFLSATU

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Perhatikan bahwa jika dilakukan substitusi = 3, 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 3 of fraction numerator square root of 3 straight x squared minus 11 end root minus square root of 3 straight x plus 7 end root over denominator square root of 6 straight x minus 9 end root minus square root of straight x squared minus 2 straight x plus 6 end root end fraction equals limit as straight x rightwards arrow 3 of open parentheses fraction numerator square root of 3 straight x squared minus 11 end root minus square root of 3 straight x plus 7 end root over denominator square root of 6 straight x minus 9 end root minus square root of straight x squared minus 2 straight x plus 6 end root end fraction times fraction numerator square root of 3 straight x squared minus 11 end root plus square root of 3 straight x plus 7 end root over denominator square root of 3 straight x squared minus 11 end root plus square root of 3 straight x plus 7 end root end fraction times fraction numerator square root of 6 straight x minus 9 end root plus square root of straight x squared minus 2 straight x plus 6 end root over denominator square root of 6 straight x minus 9 end root plus square root of straight x squared minus 2 straight x plus 6 end root end fraction close parentheses equals limit as straight x rightwards arrow 3 of fraction numerator open parentheses open parentheses 3 straight x squared minus 11 close parentheses minus open parentheses 3 straight x plus 7 close parentheses close parentheses open parentheses square root of 6 straight x minus 9 end root plus square root of straight x squared minus 2 straight x plus 6 end root close parentheses over denominator open parentheses open parentheses 6 straight x minus 9 close parentheses minus open parentheses straight x squared minus 2 straight x plus 6 close parentheses close parentheses open parentheses square root of 3 straight x squared minus 11 end root plus square root of 3 straight x plus 7 end root close parentheses end fraction equals limit as straight x rightwards arrow 3 of fraction numerator open parentheses 3 straight x squared minus 11 minus 3 straight x minus 7 close parentheses open parentheses square root of 6 straight x minus 9 end root plus square root of straight x squared minus 2 straight x plus 6 end root close parentheses over denominator open parentheses 6 straight x minus 9 minus straight x squared plus 2 straight x minus 6 close parentheses open parentheses square root of 3 straight x squared minus 11 end root plus square root of 3 straight x plus 7 end root close parentheses end fraction equals limit as straight x rightwards arrow 3 of fraction numerator open parentheses 3 straight x squared minus 3 straight x minus 18 close parentheses open parentheses square root of 6 straight x minus 9 end root plus square root of straight x squared minus 2 straight x plus 6 end root close parentheses over denominator open parentheses negative straight x squared plus 8 straight x minus 15 close parentheses open parentheses square root of 3 straight x squared minus 11 end root plus square root of 3 straight x plus 7 end root close parentheses end fraction equals limit as straight x rightwards arrow 3 of fraction numerator 3 straight x squared minus 3 straight x minus 18 over denominator negative straight x squared plus 8 straight x minus 15 end fraction times limit as straight x rightwards arrow 3 of fraction numerator square root of 6 straight x minus 9 end root plus square root of straight x squared minus 2 straight x plus 6 end root over denominator square root of 3 straight x squared minus 11 end root plus square root of 3 straight x plus 7 end root end fraction equals limit as straight x rightwards arrow 3 of fraction numerator 3 open parentheses straight x squared minus straight x minus 6 close parentheses over denominator negative open parentheses straight x squared minus 8 straight x plus 15 close parentheses end fraction times fraction numerator square root of 6 open parentheses 3 close parentheses minus 9 end root plus square root of open parentheses 3 close parentheses squared minus 2 open parentheses 3 close parentheses plus 6 end root over denominator square root of 3 open parentheses 3 close parentheses squared minus 11 end root plus square root of 3 open parentheses 3 close parentheses plus 7 end root end fraction equals limit as straight x rightwards arrow 3 of fraction numerator 3 open parentheses straight x plus 2 close parentheses open parentheses straight x minus 3 close parentheses over denominator negative open parentheses straight x minus 3 close parentheses open parentheses straight x minus 5 close parentheses end fraction times fraction numerator square root of 9 plus square root of 9 over denominator square root of 16 plus square root of 16 end fraction equals limit as straight x rightwards arrow 3 of fraction numerator 3 open parentheses straight x plus 2 close parentheses over denominator negative open parentheses straight x minus 5 close parentheses end fraction times fraction numerator 3 plus 3 over denominator 4 plus 4 end fraction equals fraction numerator 3 open parentheses 3 plus 2 close parentheses over denominator negative open parentheses 3 minus 5 close parentheses end fraction times 6 over 8 equals fraction numerator 3 open parentheses 5 close parentheses over denominator negative open parentheses negative 2 close parentheses end fraction times 6 over 8 equals 15 over 2 times 6 over 8 equals 90 over 16 equals 45 over 8 end style  

Jadi, jawaban yang tepat adalah D.

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