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Pertanyaan

Nilai dari lim x → 0 ​ 4 x + 1 ​ − 5 x + 1 ​ 2 x + 1 ​ − 3 x + 1 ​ ​ adalah ....

Nilai dari  adalah ....

  1. -begin mathsize 14px style 5 over 9 end style 

  2. -1

  3. 0

  4. 1

  5. begin mathsize 14px style 5 over 9 end style 

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I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

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Pembahasan

Perhatikan bahwa jika dilakukan substitusi x = 0 , maka didapatkan hasil . Sehingga perlu dicari cara lain untuk mencari nilai limitnya. Perhatikan bahwa

Perhatikan bahwa jika dilakukan substitusi = 0, 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 0 of fraction numerator square root of 2 straight x plus 1 end root minus square root of 3 straight x plus 1 end root over denominator square root of 4 straight x plus 1 end root minus square root of 5 straight x plus 1 end root end fraction equals limit as straight x rightwards arrow 0 of open parentheses fraction numerator square root of 2 straight x plus 1 end root minus square root of 3 straight x plus 1 end root over denominator square root of 4 straight x plus 1 end root minus square root of 5 straight x plus 1 end root end fraction times fraction numerator square root of 2 straight x plus 1 end root plus square root of 3 straight x plus 1 end root over denominator square root of 2 straight x plus 1 end root plus square root of 3 straight x plus 1 end root end fraction times fraction numerator square root of 4 straight x plus 1 end root plus square root of 5 straight x plus 1 end root over denominator square root of 4 straight x plus 1 end root plus square root of 5 straight x plus 1 end root end fraction close parentheses equals limit as straight x rightwards arrow 0 of fraction numerator open parentheses open parentheses 2 straight x plus 1 close parentheses minus open parentheses 3 straight x plus 1 close parentheses close parentheses open parentheses square root of 4 straight x plus 1 end root plus square root of 5 straight x plus 1 end root close parentheses over denominator open parentheses open parentheses 4 straight x plus 1 close parentheses minus open parentheses 5 straight x plus 1 close parentheses close parentheses open parentheses square root of 2 straight x plus 1 end root plus square root of 3 straight x plus 1 end root close parentheses end fraction equals limit as straight x rightwards arrow 0 of fraction numerator open parentheses 2 straight x plus 1 minus 3 straight x minus 1 close parentheses open parentheses square root of 4 straight x plus 1 end root plus square root of 5 straight x plus 1 end root close parentheses over denominator open parentheses 4 straight x plus 1 minus 5 straight x minus 1 close parentheses open parentheses square root of 2 straight x plus 1 end root plus square root of 3 straight x plus 1 end root close parentheses end fraction equals limit as straight x rightwards arrow 0 of fraction numerator open parentheses negative straight x close parentheses open parentheses square root of 4 straight x plus 1 end root plus square root of 5 straight x plus 1 end root close parentheses over denominator open parentheses negative straight x close parentheses open parentheses square root of 2 straight x plus 1 end root plus square root of 3 straight x plus 1 end root close parentheses end fraction equals limit as straight x rightwards arrow 0 of fraction numerator square root of 4 straight x plus 1 end root plus square root of 5 straight x plus 1 end root over denominator square root of 2 straight x plus 1 end root plus square root of 3 straight x plus 1 end root end fraction equals fraction numerator square root of 4 open parentheses 0 close parentheses plus 1 end root plus square root of 5 open parentheses 0 close parentheses plus 1 end root over denominator square root of 2 open parentheses 0 close parentheses plus 1 end root plus square root of 3 open parentheses 0 close parentheses plus 1 end root end fraction equals fraction numerator square root of 0 plus 1 end root plus square root of 0 plus 1 end root over denominator square root of 0 plus 1 end root plus square root of 0 plus 1 end root end fraction equals fraction numerator square root of 1 plus square root of 1 over denominator square root of 1 plus square root of 1 end fraction equals fraction numerator 1 plus 1 over denominator 1 plus 1 end fraction equals 2 over 2 equals 1 end style 

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