Roboguru

Nilai x→7lim​x+2​−32x2−15x+7​ adalah ....

Pertanyaan

Nilai begin mathsize 14px style limit as x rightwards arrow 7 of space fraction numerator 2 x squared minus 15 x plus 7 over denominator square root of x plus 2 end root minus 3 end fraction end style adalah ....

Pembahasan Soal:

Kita coba dengan mensubstitusikan nilai begin mathsize 14px style x equals 7 end style ke begin mathsize 14px style limit as x rightwards arrow 7 of open parentheses fraction numerator 2 x squared minus 15 x plus 7 over denominator square root of x plus 2 end root minus 3 end fraction close parentheses end style, sehingga

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 7 of open parentheses fraction numerator 2 x squared minus 15 x plus 7 over denominator square root of x plus 2 end root minus 3 end fraction close parentheses end cell equals cell fraction numerator 2 open parentheses 7 close parentheses squared minus 15 left parenthesis 7 right parenthesis plus 7 over denominator square root of 7 plus 2 end root minus 3 end fraction end cell row blank equals cell fraction numerator 98 minus 105 plus 7 over denominator 3 minus 3 end fraction end cell row blank equals cell 0 over 0 end cell end table end style 

Berdasarkan perhitungan, didapat hasil undefined adalah dalam bentuk tak tentu.

Untuk menyelesaikan soal tersebut, kita bisa coba dengan memanipulasi soal/pernyataan yang diberikan.

    begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 7 of open parentheses fraction numerator 2 x squared minus 15 x plus 7 over denominator square root of x plus 2 end root minus 3 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses fraction numerator 2 x squared minus 15 x plus 7 over denominator square root of x plus 2 end root minus 3 end fraction cross times fraction numerator square root of x plus 2 end root plus 3 over denominator square root of x plus 2 end root plus 3 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses fraction numerator open parentheses 2 x squared minus 15 x plus 7 close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses over denominator open parentheses square root of x plus 2 end root close parentheses squared minus 3 squared end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses fraction numerator open parentheses 2 x squared minus 15 x plus 7 close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses over denominator x plus 2 minus 9 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses x minus 7 close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses over denominator x minus 7 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses up diagonal strike x minus 7 end strike close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses over denominator up diagonal strike x minus 7 end strike end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses fraction numerator open parentheses 2 x minus 1 close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses over denominator 1 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 7 of open parentheses open parentheses 2 x minus 1 close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses close parentheses end cell end table end style 

Setelah kita manipulasi dan dapat bentuk paling sederhana, kita substitusikan nilai undefined.

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 7 of open parentheses fraction numerator 2 x squared minus 15 x plus 7 over denominator square root of x plus 2 end root minus 3 end fraction close parentheses end cell equals cell limit as x rightwards arrow 7 of open parentheses open parentheses 2 x minus 1 close parentheses open parentheses square root of x plus 2 end root plus 3 close parentheses close parentheses end cell row blank equals cell open parentheses open parentheses 2 open parentheses 7 close parentheses minus 1 close parentheses open parentheses square root of 7 plus 2 end root plus 3 close parentheses close parentheses end cell row blank equals cell open parentheses 14 minus 1 close parentheses open parentheses square root of 9 plus 3 close parentheses end cell row blank equals cell open parentheses 13 close parentheses open parentheses 3 plus 3 close parentheses end cell row blank equals cell open parentheses 13 close parentheses open parentheses 6 close parentheses end cell row blank equals 78 end table end style 

Jadi, nilai dari begin mathsize 14px style stack bold l bold i bold m with bold x bold rightwards arrow bold 7 below bold space fraction numerator bold 2 bold x to the power of bold 2 bold minus bold 15 bold x bold plus bold 7 over denominator square root of bold x bold plus bold 2 end root bold minus bold 3 end fraction end style adalah begin mathsize 14px style bold 78 end style.undefined

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. RGFREELANCETIGASATU

Terakhir diupdate 05 Oktober 2021

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Pertanyaan yang serupa

x→1lim​2+2x​−6−2x​x3−x2​=

Pembahasan Soal:

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0

Roboguru

Untuk soal nomor 7-13, tentukan harga-harga limitnya dengan terlebih dahulu mengalikannya dengan bentuk sekawan. x→1lim​1−x2x2+3​−x−1​

Pembahasan Soal:

Dengan terlebih dahulu mengalikannya dengan bentuk sekawan, dapat diperoleh

begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator square root of x squared plus 3 end root minus x minus 1 over denominator 1 minus x squared end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator square root of x squared plus 3 end root minus left parenthesis x plus 1 right parenthesis over denominator 1 minus x squared end fraction end cell row blank equals cell limit as x rightwards arrow 1 of open parentheses fraction numerator square root of x squared plus 3 end root minus left parenthesis x plus 1 right parenthesis over denominator 1 minus x squared end fraction cross times fraction numerator square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis over denominator square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator x squared plus 3 minus left parenthesis x plus 1 right parenthesis squared over denominator open parentheses 1 minus x squared close parentheses cross times open parentheses square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator x squared plus 3 minus left parenthesis x squared plus 2 x plus 1 right parenthesis over denominator open parentheses 1 minus x squared close parentheses cross times open parentheses square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator x squared plus 3 minus x squared minus 2 x minus 1 over denominator open parentheses 1 minus x squared close parentheses cross times open parentheses square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 2 minus 2 x over denominator open parentheses 1 minus x squared close parentheses cross times open parentheses square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 2 cross times open parentheses 1 minus x close parentheses over denominator open parentheses 1 minus x close parentheses cross times open parentheses 1 plus x close parentheses cross times open parentheses square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 2 over denominator open parentheses 1 plus x close parentheses cross times open parentheses square root of x squared plus 3 end root plus left parenthesis x plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator open parentheses 1 plus 1 close parentheses cross times open parentheses square root of 1 squared plus 3 end root plus left parenthesis 1 plus 1 right parenthesis close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator 2 cross times open parentheses square root of 1 plus 3 end root plus 2 close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator 2 cross times open parentheses square root of 4 plus 2 close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator 2 cross times open parentheses 2 plus 2 close parentheses end fraction end cell row blank equals cell fraction numerator 2 over denominator 2 cross times 4 end fraction end cell row blank equals cell 1 fourth end cell end table end style  

Jadi, diperoleh nilai limit di atas adalah begin mathsize 14px style bold 1 over bold 4 end style 

0

Roboguru

x→4lim​x2−2x−83−x2−7​​=...

Pembahasan Soal:

Jika disubstitusikan nilai begin mathsize 14px style x equals 3 end style, diperoleh bentuk tak tentu begin mathsize 14px style 0 over 0 end style. Sehingga nilai limitnya dapat ditentukan dengan kali sekawan,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow 4 of fraction numerator 3 minus square root of straight x squared minus 7 end root over denominator straight x squared minus 2 straight x minus 8 end fraction end cell equals cell limit as straight x rightwards arrow 4 of fraction numerator 3 minus square root of straight x squared minus 7 end root over denominator straight x squared minus 2 straight x minus 8 end fraction cross times fraction numerator 3 plus square root of straight x squared minus 7 end root over denominator 3 plus square root of straight x squared minus 7 end root end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator 9 minus open parentheses straight x squared minus 7 close parentheses over denominator open parentheses straight x squared minus 2 straight x minus 8 close parentheses open parentheses 3 plus square root of straight x squared minus 7 end root close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator 16 minus straight x squared over denominator left parenthesis straight x squared minus 2 straight x minus 8 right parenthesis left parenthesis 3 plus square root of straight x squared minus 7 end root right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator open parentheses 4 minus straight x close parentheses open parentheses 4 plus straight x close parentheses over denominator left parenthesis straight x minus 4 right parenthesis open parentheses straight x plus 2 close parentheses left parenthesis 3 plus square root of straight x squared minus 7 end root right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator negative left parenthesis 4 plus straight x right parenthesis over denominator open parentheses straight x plus 2 close parentheses left parenthesis 3 plus square root of straight x squared minus 7 end root right parenthesis end fraction end cell row blank equals cell fraction numerator negative left parenthesis 4 plus 4 right parenthesis over denominator open parentheses 4 plus 2 close parentheses left parenthesis 3 plus square root of 4 squared minus 7 end root right parenthesis end fraction end cell row blank equals cell fraction numerator negative 8 over denominator 6 left parenthesis 6 right parenthesis end fraction end cell row blank equals cell negative 2 over 9 end cell end table end style

Jadi, hasil limitnya adalah begin mathsize 14px style negative 2 over 9 end style.

0

Roboguru

Tentukan nilai limit dari fungsi-fungsi berikut ini: x→4lim​5−x2+9​16−x2​

Pembahasan Soal:

Dengan menggunakan metode mengalikan akar sekawan, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator 16 minus x squared over denominator 5 minus square root of x squared plus 9 end root end fraction end cell equals cell limit as x rightwards arrow 4 of open parentheses fraction numerator 16 minus x squared over denominator 5 minus square root of x squared plus 9 end root end fraction times fraction numerator 5 plus square root of x squared plus 9 end root over denominator 5 plus square root of x squared plus 9 end root end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses 16 minus x squared close parentheses open parentheses 5 plus square root of x squared plus 9 end root close parentheses over denominator open parentheses 5 minus square root of x squared plus 9 end root close parentheses open parentheses 5 plus square root of x squared plus 9 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses 16 minus x squared close parentheses open parentheses 5 plus square root of x squared plus 9 end root close parentheses over denominator 25 minus open parentheses x squared plus 9 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses 16 minus x squared close parentheses open parentheses 5 plus square root of x squared plus 9 end root close parentheses over denominator 25 minus x squared minus 9 end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator down diagonal strike open parentheses 16 minus x squared close parentheses end strike open parentheses 5 plus square root of x squared plus 9 end root close parentheses over denominator down diagonal strike 16 minus x squared end strike end fraction end cell row blank equals cell limit as x rightwards arrow 4 of open parentheses 5 plus square root of x squared plus 9 end root close parentheses end cell row blank equals cell 5 plus square root of 4 squared plus 9 end root end cell row blank equals cell 5 plus square root of 16 plus 9 end root end cell row blank equals cell 5 plus square root of 25 end cell row blank equals cell 5 plus 5 end cell row blank equals 10 end table end style 

Jadi, nilai dari begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator 16 minus x squared over denominator 5 minus square root of x squared plus 9 end root end fraction end style adalah begin mathsize 14px style 10 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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