Pertanyaan

x → 2 lim 4 x + 1 − 3 7 x + 2 − 6 x + 4 = ...

$x \to 2 lim \frac{7 x + 2 - 6 x + 4}{4 x + 1 - 3} =$ ...

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Klaim

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Dengan menggunakan metode mengkalikan akar sekawan: Karena limit tersebut tidak dapat dihitungdan ekspresi tersebut tidak dapat di sederhanakan, sehingga menghitungnya dapat menggunakan limit kiri dan limit kanan. Didapat: Karena , sehingga . Jadi, . Dengan demikian, jawaban yang tepat adalah C.

Dengan menggunakan metode mengkalikan akar sekawan:

$begin mathsize 14px style limit as x rightwards arrow 2 of open parentheses fraction numerator square root of 7 straight x plus 2 end root minus square root of 6 straight x plus 4 end root over denominator square root of 4 straight x plus 1 end root minus 3 end fraction close parentheses equals limit as x rightwards arrow 2 of open parentheses fraction numerator square root of 7 straight x plus 2 end root minus square root of 6 straight x plus 4 end root over denominator square root of 4 straight x plus 1 end root minus 3 end fraction times fraction numerator square root of 4 x plus 1 end root plus 3 over denominator square root of 4 x plus 1 end root plus 3 end fraction close parentheses equals limit as x rightwards arrow 2 of open parentheses fraction numerator open parentheses square root of 7 straight x plus 2 end root minus square root of 6 straight x plus 4 end root close parentheses open parentheses square root of 4 x plus 1 end root plus 3 close parentheses over denominator 4 straight x plus 1 minus 9 end fraction close parentheses equals limit as x rightwards arrow 2 of open parentheses fraction numerator open parentheses square root of 7 straight x plus 2 end root minus square root of 6 straight x plus 4 end root close parentheses open parentheses square root of 4 x plus 1 end root plus 3 close parentheses over denominator 4 straight x minus 8 end fraction close parentheses end style$

Karena limit tersebut tidak dapat dihitung dan ekspresi tersebut tidak dapat di sederhanakan, sehingga menghitungnya dapat menggunakan limit kiri dan limit kanan. Didapat:

$begin mathsize 14px style limit as x rightwards arrow 2 to the power of minus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses open parentheses square root of 4 x plus 1 end root plus 3 close parentheses over denominator 4 x minus 8 end fraction close parenthesesequals limit as x rightwards arrow 2 to the power of minus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses over denominator 4 x minus 8 end fraction times open parentheses square root of 4 x plus 1 end root plus 3 close parentheses close parenthesesequals limit as x rightwards arrow 2 to the power of minus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses over denominator 4 x minus 8 end fraction close parentheses times limit as x rightwards arrow 2 to the power of minus of open parentheses square root of 4 x plus 1 end root plus 3 close parenthesesequals limit as x rightwards arrow 2 to the power of minus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses times begin display style fraction numerator open parentheses square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root close parentheses over denominator open parentheses square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root close parentheses end fraction end style over denominator 4 x minus 8 end fraction close parentheses times limit as x rightwards arrow 2 to the power of minus of open parentheses square root of 4 x plus 1 end root plus 3 close parenthesesequals limit as x rightwards arrow thin space 2 to the power of minus of open parentheses fraction numerator fraction numerator x minus 2 over denominator square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root end fraction over denominator 4 x minus 8 end fraction close parentheses times open parentheses square root of 4 left parenthesis 2 right parenthesis plus 1 end root plus 3 close parenthesesequals limit as x rightwards arrow thin space 2 to the power of minus of open parentheses fraction numerator x minus 2 over denominator square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root times 4 open parentheses x minus 2 close parentheses end fraction close parentheses open parentheses square root of 9 plus 3 close parenthesesequals fraction numerator 1 over denominator 4 open parentheses square root of 7 times thin space 2 plus 2 end root plus square root of 6 times thin space 2 plus 4 end root close parentheses end fraction open parentheses 3 plus 3 close parenthesesequals fraction numerator 1 over denominator 4 open parentheses square root of 16 plus square root of 16 close parentheses end fraction left parenthesis 6 right parenthesisequals fraction numerator 1 over denominator 4 open parentheses 8 close parentheses end fraction left parenthesis 6 right parenthesisequals 3 over 16limit as x rightwards arrow 2 to the power of plus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses open parentheses square root of 4 x plus 1 end root plus 3 close parentheses over denominator 4 x minus 8 end fraction close parenthesesequals limit as x rightwards arrow 2 to the power of plus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses over denominator 4 x minus 8 end fraction times open parentheses square root of 4 x plus 1 end root plus 3 close parentheses close parenthesesequals limit as x rightwards arrow 2 to the power of plus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses over denominator 4 x minus 8 end fraction close parentheses times limit as x rightwards arrow 2 to the power of plus of open parentheses square root of 4 x plus 1 end root plus 3 close parenthesesequals limit as x rightwards arrow 2 to the power of plus of open parentheses fraction numerator open parentheses square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root close parentheses times begin display style fraction numerator open parentheses square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root close parentheses over denominator open parentheses square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root close parentheses end fraction end style over denominator 4 x minus 8 end fraction close parentheses times limit as x rightwards arrow 2 to the power of plus of open parentheses square root of 4 x plus 1 end root plus 3 close parenthesesequals limit as x rightwards arrow thin space 2 to the power of plus of open parentheses fraction numerator fraction numerator x minus 2 over denominator square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root end fraction over denominator 4 x minus 8 end fraction close parentheses times open parentheses square root of 4 left parenthesis 2 right parenthesis plus 1 end root plus 3 close parenthesesequals limit as x rightwards arrow thin space 2 to the power of plus of open parentheses fraction numerator x minus 2 over denominator square root of 7 x plus 2 end root plus square root of 6 x plus 4 end root times 4 open parentheses x minus 2 close parentheses end fraction close parentheses open parentheses square root of 9 plus 3 close parenthesesequals fraction numerator 1 over denominator 4 open parentheses square root of 7 times thin space 2 plus 2 end root plus square root of 6 times thin space 2 plus 4 end root close parentheses end fraction open parentheses 3 plus 3 close parenthesesequals fraction numerator 1 over denominator 4 open parentheses square root of 16 plus square root of 16 close parentheses end fraction left parenthesis 6 right parenthesisequals fraction numerator 1 over denominator 4 open parentheses 8 close parentheses end fraction left parenthesis 6 right parenthesisequals 3 over 16 end style$

Karena , sehingga .

Jadi, $begin mathsize 14px style limit as x rightwards arrow 2 of fraction numerator square root of 7 x plus 2 end root minus square root of 6 x plus 4 end root over denominator square root of 4 x plus 1 end root minus 3 end fraction equals 3 over 16 end style$ .

Dengan demikian, jawaban yang tepat adalah C.