Roboguru

Dengan merasionalkan penyebut, carilah nilai limit dari: (b)

Pertanyaan

Dengan merasionalkan penyebut, carilah nilai limit dari:

(b)  limit as x rightwards arrow 3 of space fraction numerator 9 minus x squared over denominator 4 minus square root of x squared plus 7 end root end fraction

Pembahasan Soal:

Dengan mengalikan akar sekawan dari penyebut diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 3 of space fraction numerator 9 minus x squared over denominator 4 minus square root of x squared plus 7 end root end fraction end cell equals cell limit as x rightwards arrow 3 of space fraction numerator 9 minus x squared over denominator 4 minus square root of x squared plus 7 end root end fraction cross times fraction numerator 4 plus square root of x squared plus 7 end root over denominator 4 plus square root of x squared plus 7 end root end fraction end cell row blank equals cell limit as x rightwards arrow 3 of space fraction numerator open parentheses 9 minus x squared close parentheses open parentheses 4 plus square root of x squared plus 7 end root close parentheses over denominator 16 minus open parentheses x squared plus 7 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 3 of space fraction numerator open parentheses 9 minus x squared close parentheses open parentheses 4 plus square root of x squared plus 7 end root close parentheses over denominator 9 minus x squared end fraction end cell row blank equals cell limit as x rightwards arrow 3 of space open parentheses 4 plus square root of x squared plus 7 end root close parentheses end cell row blank equals cell 4 plus square root of 3 squared plus 7 end root end cell row blank equals cell 4 plus square root of 16 end cell row blank equals cell 4 plus 4 end cell row blank equals 16 end table

Dengan demikian nilai dari limit as x rightwards arrow 3 of space fraction numerator 9 minus x squared over denominator 4 minus square root of x squared plus 7 end root end fraction equals 16.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Nur

Mahasiswa/Alumni Institut Teknologi Sepuluh Nopember

Terakhir diupdate 02 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Nilai dari

Pembahasan Soal:

Pada soal ini, metode yang sesuai adalah dengan pengalian faktor sekawan karena apabila dikerjakan dengan metode substitusi akan menghasilkan bilangan tak hingga.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator square root of 2 x plus 1 end root minus square root of x plus 5 end root over denominator square root of 3 x plus 4 end root minus square root of 5 x minus 4 end root end fraction end cell equals cell limit as x rightwards arrow 4 of fraction numerator square root of 2 x plus 1 end root minus square root of x plus 5 end root over denominator square root of 3 x plus 4 end root minus square root of 5 x minus 4 end root end fraction times fraction numerator square root of 2 x plus 1 end root plus square root of x plus 5 end root over denominator square root of 2 x plus 1 end root plus square root of x plus 5 end root end fraction times fraction numerator square root of 3 x plus 4 end root plus square root of 5 x minus 4 end root over denominator square root of 3 x plus 4 end root plus square root of 5 x minus 4 end root end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses 2 x plus 1 minus x plus 5 close parentheses times square root of 3 x plus 4 end root plus square root of 5 x minus 4 end root over denominator open parentheses 3 x plus 4 minus 5 x minus 4 close parentheses times square root of 2 x plus 1 end root plus square root of x plus 5 end root end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator open parentheses x plus 6 close parentheses times square root of 3 x plus 4 end root plus square root of 5 x minus 4 end root over denominator open parentheses negative 2 x close parentheses times square root of 2 x plus 1 end root plus square root of x plus 5 end root end fraction end cell row blank equals cell fraction numerator open parentheses 4 plus 6 close parentheses times square root of 3 open parentheses 4 close parentheses plus 4 end root plus square root of 5 open parentheses 4 close parentheses minus 4 end root over denominator open parentheses negative 2 open parentheses 4 close parentheses close parentheses times square root of 2 open parentheses 4 close parentheses plus 1 end root plus square root of open parentheses 4 close parentheses plus 5 end root end fraction end cell row blank equals cell fraction numerator 10 open parentheses square root of 12 plus 4 end root plus square root of 20 minus 4 end root close parentheses over denominator negative 8 open parentheses square root of 8 plus 1 end root plus square root of 4 plus 5 end root close parentheses end fraction end cell row blank equals cell fraction numerator 10 open parentheses square root of 16 plus square root of 16 close parentheses over denominator negative 8 open parentheses square root of 9 plus square root of 9 close parentheses end fraction end cell row blank equals cell fraction numerator 10 open parentheses 4 plus 4 close parentheses over denominator negative 8 open parentheses 3 plus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 10 open parentheses 8 close parentheses over denominator negative 8 open parentheses 6 close parentheses end fraction end cell row blank equals cell negative 80 over 48 end cell row blank equals cell negative 5 over 3 end cell end table 

Jadi, nilai dari limit tersebut adalah negative 5 over 3 

0

Roboguru

Hitunglah tiap limit fungsi berikut!

Pembahasan Soal:

Gunakan akar sekawan untuk mengerjakan soal berikut
Akar sekawan dari begin mathsize 14px style 2 minus square root of x squared plus 3 end root end style adalah begin mathsize 14px style 2 plus square root of italic x squared plus 3 end root end style 

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

Jadi, diperoleh hasil begin mathsize 14px style lim with italic x yields 1 below fraction numerator 1 minus sign italic x squared over denominator 2 minus sign square root of italic x squared plus 3 end root end fraction equals 4 end style 

0

Roboguru

Nilai dari

Pembahasan Soal:

Dengan mengalikan akar sekawan.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of fraction numerator square root of 3 x plus 6 end root minus 3 over denominator 2 minus square root of 3 x plus 1 end root end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator square root of 3 x plus 6 end root minus 3 over denominator 2 minus square root of 3 x plus 1 end root end fraction times fraction numerator 2 plus square root of 3 x plus 1 end root over denominator 2 plus square root of 3 x plus 1 end root end fraction times fraction numerator square root of 3 x plus 6 end root plus 3 over denominator square root of 3 x plus 6 end root plus 3 end fraction space end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator open parentheses square root of 3 x plus 6 end root close parentheses squared minus 3 squared over denominator 2 squared minus open parentheses square root of 3 x plus 1 end root close parentheses squared end fraction times fraction numerator 2 plus square root of 3 x plus 1 end root over denominator square root of 3 x plus 6 end root plus 3 end fraction end cell row space equals cell limit as x rightwards arrow 1 of fraction numerator 3 x plus 6 minus 9 over denominator 4 minus open parentheses 3 x plus 1 close parentheses end fraction times fraction numerator 2 plus square root of 3 x plus 1 end root over denominator square root of 3 x plus 6 end root plus 3 end fraction space end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 3 x minus 3 over denominator 3 minus 3 x end fraction times fraction numerator 2 plus square root of 3 x plus 1 end root over denominator square root of 3 x plus 6 end root plus 3 end fraction space end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative up diagonal strike open parentheses 3 minus 3 x close parentheses end strike over denominator up diagonal strike open parentheses 3 minus 3 x close parentheses end strike end fraction times fraction numerator 2 plus square root of 3 x plus 1 end root over denominator square root of 3 x plus 6 end root plus 3 end fraction space end cell row blank equals cell limit as x rightwards arrow 1 of minus fraction numerator 2 plus square root of 3 x plus 1 end root over denominator square root of 3 x plus 6 end root plus 3 end fraction end cell row space equals cell negative fraction numerator 2 plus square root of 3 times 1 plus 1 end root over denominator square root of 3 times 1 plus 6 end root plus 3 end fraction end cell row space equals cell negative fraction numerator 2 plus square root of 4 over denominator square root of 9 plus 3 end fraction end cell row space equals cell negative fraction numerator 2 plus 2 over denominator 3 plus 3 end fraction space end cell row blank equals cell negative 4 over 6 space end cell row blank equals cell negative 2 over 3 end cell row blank equals cell fraction numerator negative 2 over denominator 3 end fraction end cell end table end style    

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Tentukan nilai limit dari fungsi-fungsi berikut ini:

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

Jika  ....

Pembahasan Soal:

Perhatikan bahwa

begin mathsize 14px style limit as straight x rightwards arrow 3 of fraction numerator straight f open parentheses straight x close parentheses over denominator square root of 1 plus straight x end root minus 2 end fraction equals limit as straight x rightwards arrow 3 of open parentheses fraction numerator straight f open parentheses straight x close parentheses over denominator square root of 1 plus straight x end root minus 2 end fraction times fraction numerator square root of 1 plus straight x end root plus 2 over denominator square root of 1 plus straight x end root plus 2 end fraction close parentheses equals limit as straight x rightwards arrow 3 of fraction numerator straight f open parentheses straight x close parentheses open parentheses square root of 1 plus straight x end root plus 2 close parentheses over denominator open parentheses square root of 1 plus straight x end root close parentheses squared minus 2 squared end fraction equals limit as straight x rightwards arrow 3 of fraction numerator straight f open parentheses straight x close parentheses open parentheses square root of 1 plus straight x end root plus 2 close parentheses over denominator open parentheses 1 plus straight x close parentheses minus 4 end fraction equals limit as straight x rightwards arrow 3 of fraction numerator straight f open parentheses straight x close parentheses open parentheses square root of 1 plus straight x end root plus 2 close parentheses over denominator straight x minus 3 end fraction equals limit as straight x rightwards arrow 3 of fraction numerator straight f open parentheses straight x close parentheses over denominator straight x minus 3 end fraction times limit as straight x rightwards arrow 3 of open parentheses square root of 1 plus straight x end root plus 2 close parentheses equals 5 open parentheses square root of 1 plus 3 end root plus 2 close parentheses equals 5 open parentheses square root of 4 plus 2 close parentheses equals 5 open parentheses 2 plus 2 close parentheses equals 5 open parentheses 4 close parentheses equals 20 end style 

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

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