Roboguru

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

Pertanyaan

Dengan merasionalkan penyebut, carilah nilai limit dari:

(a)  limit as x rightwards arrow 1 of space fraction numerator x squared minus 1 over denominator square root of 3 x squared minus 3 end root end fraction 

Pembahasan Soal:

Dengan merasionalkan penyebut diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of space fraction numerator x squared minus 1 over denominator square root of 3 x squared minus 3 end root end fraction end cell equals cell limit as x rightwards arrow 1 of space fraction numerator x squared minus 1 over denominator square root of 3 x squared minus 3 end root end fraction cross times fraction numerator square root of 3 x squared minus 3 end root over denominator square root of 3 x squared minus 3 end root end fraction end cell row blank equals cell limit as x rightwards arrow 1 of space fraction numerator open parentheses x squared minus 1 close parentheses open parentheses square root of 3 x squared minus 3 end root close parentheses over denominator 3 x squared minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 1 of space fraction numerator open parentheses x squared minus 1 close parentheses open parentheses square root of 3 x squared minus 3 end root close parentheses over denominator 3 open parentheses x squared minus 1 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of space fraction numerator open parentheses square root of 3 x squared minus 3 end root close parentheses over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses square root of 3 open parentheses 1 close parentheses squared minus 3 end root close parentheses over denominator 3 end fraction end cell row blank equals cell fraction numerator square root of 3 minus 3 end root over denominator 3 end fraction end cell row blank equals cell fraction numerator square root of 0 over denominator 3 end fraction end cell row blank equals cell 0 over 3 end cell row blank equals 0 end table

Dengan demikian nilai dari table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x squared minus 1 over denominator square root of 3 x squared minus 3 end root 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 0 end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Nur

Mahasiswa/Alumni Institut Teknologi Sepuluh Nopember

Terakhir diupdate 02 Mei 2021

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Pertanyaan yang serupa

Nilai  adalah ....

Pembahasan Soal:

Dengan mengalikan akar sekawan.

begin mathsize 14px style limit as x rightwards arrow 4 of fraction numerator 4 minus x over denominator square root of 12 minus square root of x plus 8 end root end fraction equals limit as x rightwards arrow 4 of fraction numerator 4 minus x over denominator square root of 12 minus square root of x plus 8 end root end fraction times fraction numerator square root of 12 plus square root of x plus 8 end root over denominator square root of 12 plus square root of x plus 8 end root end fraction equals limit as x rightwards arrow 4 of fraction numerator open parentheses 4 minus x close parentheses open parentheses square root of 12 plus square root of x plus 8 end root close parentheses over denominator 12 minus open parentheses x plus 8 close parentheses end fraction equals limit as x rightwards arrow 4 of fraction numerator up diagonal strike open parentheses 4 minus x close parentheses end strike open parentheses square root of 12 plus square root of x plus 8 end root close parentheses over denominator up diagonal strike open parentheses 4 minus x close parentheses end strike end fraction equals limit as x rightwards arrow 4 of square root of 12 plus square root of x plus 8 end root equals square root of 12 plus square root of 4 plus 8 end root equals square root of 12 plus square root of 12 equals 2 square root of 12 equals 2 square root of 4 times 3 end root equals 2 times 2 square root of 3 equals 4 square root of 3 end style 

Jadi, jawaban yang tepat adalah E.

0

Roboguru

Nilai dari

Pembahasan Soal:

Nilai limit tersebut dapat diketahui dengan mengalikan akar sekawan pembilangnya. Sehingga :

limit as x rightwards arrow 2 of fraction numerator square root of 3 x minus 4 end root minus square root of x over denominator x minus 2 end fraction cross times fraction numerator square root of 3 x minus 4 end root plus square root of x over denominator square root of 3 x minus 4 end root plus square root of x end fraction equals limit as x rightwards arrow 2 of fraction numerator 3 x minus 4 minus x over denominator open parentheses x minus 2 close parentheses open parentheses square root of 3 x minus 4 end root plus square root of x close parentheses end fraction equals limit as x rightwards arrow 2 of fraction numerator 2 x minus 4 over denominator open parentheses x minus 2 close parentheses open parentheses square root of 3 x minus 4 end root plus square root of x close parentheses end fraction equals limit as x rightwards arrow 2 of fraction numerator 2 up diagonal strike open parentheses x minus 2 close parentheses end strike over denominator up diagonal strike open parentheses x minus 2 close parentheses end strike open parentheses square root of 3 x minus 4 end root plus square root of x close parentheses end fraction equals limit as x rightwards arrow 2 of fraction numerator 2 over denominator square root of 3 x minus 4 end root plus square root of x end fraction equals fraction numerator 2 over denominator square root of 3 left parenthesis 2 right parenthesis minus 4 end root plus square root of 2 end fraction equals fraction numerator 2 over denominator square root of 2 plus square root of 2 end fraction equals fraction numerator 2 over denominator 2 square root of 2 end fraction cross times fraction numerator square root of 2 over denominator square root of 2 end fraction equals fraction numerator 2 square root of 2 over denominator 4 end fraction equals 1 half square root of 2 

Maka, nilai dari limit as x rightwards arrow 2 of fraction numerator square root of 3 x minus 4 end root minus square root of x over denominator x minus 2 end fraction equals 1 half square root of 2.

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 blank blank cell limit as x rightwards arrow 0 of fraction numerator 5 x over denominator square root of x squared minus 2 x plus 1 end root minus square root of x squared plus 3 x plus 1 end root end fraction end cell row blank equals cell limit as x rightwards arrow 0 of open parentheses fraction numerator 5 x over denominator square root of x squared minus 2 x plus 1 end root minus square root of x squared plus 3 x plus 1 end root end fraction times fraction numerator square root of x squared minus 2 x plus 1 end root plus square root of x squared plus 3 x plus 1 end root over denominator square root of x squared minus 2 x plus 1 end root plus square root of x squared plus 3 x plus 1 end root end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses 5 x close parentheses open parentheses square root of x squared minus 2 x plus 1 end root plus square root of x squared plus 3 x plus 1 end root close parentheses over denominator open parentheses x squared minus 2 x plus 1 close parentheses minus open parentheses x squared plus 3 x plus 1 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses 5 x close parentheses open parentheses square root of x squared minus 2 x plus 1 end root plus square root of x squared plus 3 x plus 1 end root close parentheses over denominator x squared minus 2 x plus 1 minus x squared minus 3 x minus 1 end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses 5 x close parentheses open parentheses square root of x squared minus 2 x plus 1 end root plus square root of x squared plus 3 x plus 1 end root close parentheses over denominator negative 5 x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator open parentheses square root of x squared minus 2 x plus 1 end root plus square root of x squared plus 3 x plus 1 end root close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator square root of 0 squared minus 2 open parentheses 0 close parentheses plus 1 end root plus square root of 0 squared plus 3 open parentheses 0 close parentheses plus 1 end root over denominator negative 1 end fraction end cell row blank equals cell fraction numerator square root of 1 plus square root of 1 over denominator negative 1 end fraction end cell row blank equals cell fraction numerator 1 plus 1 over denominator negative 1 end fraction end cell row blank equals cell fraction numerator 2 over denominator negative 1 end fraction end cell row blank equals cell negative 2 end cell end table end style 

Jadi nilai dari begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator 5 x over denominator square root of x squared minus 2 x plus 1 end root minus square root of x squared plus 3 x plus 1 end root end fraction end style adalah begin mathsize 14px style negative 2 end style

0

Roboguru

Tentukan nilai limit fungsi berikut. a.

Pembahasan Soal:

begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator x over denominator 3 minus square root of 9 minus x end root end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator x over denominator 3 minus square root of 9 minus x end root end fraction. fraction numerator 3 plus square root of 9 minus x end root over denominator 3 plus square root of 9 minus x end root end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator x open parentheses 3 plus square root of 9 minus x end root close parentheses over denominator 9 minus open parentheses 9 minus x close parentheses end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator x left parenthesis 3 plus square root of 9 minus x end root right parenthesis over denominator x end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses 3 plus square root of 9 minus x end root close parentheses space space space space space space space space space space space space space space space space space space space space space space space space equals 3 plus square root of 9 minus 0 end root space space space space space space space space space space space space space space space space space space space space space space space space equals 3 plus 3 space space space space space space space space space space space space space space space space space space space space space space space space equals 6 end style 

Jadi,  begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator x over denominator 3 minus square root of 9 minus x end root end fraction equals 6 end style.

0

Roboguru

Find the value of the limit. a.

Pembahasan Soal:

limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals fraction numerator square root of 0 plus 2 end root minus square root of 2 over denominator 0 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator square root of 2 minus square root of 2 over denominator 0 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 0 over 0 

Karena bentuk limit tersebut merupakan bentuk limit tak tentu, maka kita gunakan kali akar sekawan. Seperti berikut ini,

limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals limit as x rightwards arrow 0 of open parentheses fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction. fraction numerator square root of x plus 2 end root plus square root of 2 over denominator square root of x plus 2 end root plus square root of 2 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator open parentheses x plus 2 close parentheses minus 2 over denominator x open parentheses square root of x plus 2 end root plus square root of 2 close parentheses end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator up diagonal strike x over denominator up diagonal strike x open parentheses square root of x plus 2 end root plus square root of 2 close parentheses end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals limit as x rightwards arrow 0 of open parentheses fraction numerator 1 over denominator square root of x plus 2 end root plus square root of 2 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator square root of 0 plus 2 end root plus square root of 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 2 square root of 2 end fraction. fraction numerator square root of 2 over denominator square root of 2 end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 fourth square root of 2

Jadi, limit as x rightwards arrow 0 of fraction numerator square root of x plus 2 end root minus square root of 2 over denominator x end fraction equals 1 fourth square root of 2.

0

Roboguru

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