Roboguru

Jika x→0lim​xAx+B​​=1, maka ....

Pertanyaan

Jika begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator square root of A x plus B end root over denominator x end fraction equals 1 end style, maka ....

  1. begin mathsize 14px style straight B equals straight A squared end style 

  2. begin mathsize 14px style 3 straight B squared equals straight A end style 

  3. begin mathsize 14px style 4 straight B equals straight A squared end style 

  4. begin mathsize 14px style 4 straight B equals straight A end style  

  5. begin mathsize 14px style straight A plus straight B equals 0 end style  

Pembahasan Soal:

begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator square root of A x plus B end root over denominator x end fraction equals 1 end style

menggunakan Dalil L'Hospital

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

Jadi, jawaban yang tepat adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Mustikowati

Mahasiswa/Alumni Universitas Negeri Jakarta

Terakhir diupdate 05 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

x→1lim​2+2x​−6−2x​x3−x2​=

Pembahasan Soal:

undefined

0

Roboguru

x→−3lim​x−3x2+9​=...

Pembahasan Soal:

Perhatikan perhitungan berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow negative 3 of fraction numerator x squared plus 9 over denominator x minus 3 end fraction end cell equals cell fraction numerator left parenthesis negative 3 right parenthesis squared plus 9 over denominator negative 3 minus 3 end fraction end cell row blank equals cell fraction numerator 9 plus 9 over denominator negative 6 end fraction end cell row blank equals cell fraction numerator 18 over denominator negative 6 end fraction end cell row blank equals cell negative 3 end cell end table end style 

Jadi, diperoleh begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow negative 3 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x squared plus 9 over denominator x minus 3 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table end style.

0

Roboguru

x→−2lim​(x2+x−2x3+2x2​)=...

Pembahasan Soal:

Soal limit diatas dapat diselesaikan dengan mefaktorkan pembilang dan penyebut, kemudian membagi pembilang dan penyebut, sebagai berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow negative 2 of open parentheses fraction numerator x cubed plus 2 x squared over denominator x squared plus x minus 2 end fraction close parentheses end cell equals cell limit as x rightwards arrow negative 2 of open parentheses fraction numerator x squared open parentheses x plus 2 close parentheses over denominator open parentheses x plus 2 close parentheses open parentheses x minus 1 close parentheses end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow negative 2 of open parentheses fraction numerator x squared over denominator open parentheses x minus 1 close parentheses end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses negative 2 close parentheses squared over denominator open parentheses negative 2 minus 1 close parentheses end fraction end cell row blank equals cell negative 4 over 3 end cell end table end style 

Sehingga, begin mathsize 14px style limit as x rightwards arrow negative 2 of open parentheses fraction numerator x cubed plus 2 x squared over denominator x squared plus x minus 2 end fraction close parentheses equals negative 4 over 3 end style.

0

Roboguru

Nilai dari x→1lim​x−35x2​=…

Pembahasan Soal:

Limit bentuk tertentu dengan substitusi nilai x.

begin mathsize 14px style limit as x rightwards arrow 1 of fraction numerator 5 x squared over denominator x minus 3 end fraction equals fraction numerator 5 times 1 squared over denominator 1 minus 3 end fraction equals fraction numerator 5 over denominator negative 2 end fraction equals negative 5 over 2 end style      

Jadi, hasil dari limit tersebut adalah begin mathsize 14px style negative 5 over 2 end style.

0

Roboguru

limx→−1​x2−12−x2​−42−x4​​= ....

Pembahasan Soal:

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.

0

Roboguru

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