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, maka ....

Pertanyaan

begin mathsize 14px style f left parenthesis x right parenthesis space equals space open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell x squared minus 8 x plus 3 comma space end cell cell x space greater than space 7 end cell row cell 17 minus 3 x end cell cell x less than 7 end cell end table close end style, maka begin mathsize 14px style limit as x rightwards arrow negative 3 of f left parenthesis x right parenthesis equals end style....

  1. 26

  2. 36

  3. 8

  4. 16

  5. 28

Pembahasan Soal:

Perhatikan bahwa x3 kurang dari 7 maka fungsi yang digunakan adalah f(x)=173x. Sehingga limit fungsi diperoleh:

limx3f(x)====limx3173x173(3)17+926 

Dengan demikian, nilai dari limx3f(x)=26.

Oleh karena itu, jawaban yang benar adalah A.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

P. Tessalonika

Mahasiswa/Alumni Universitas Negeri Medan

Terakhir diupdate 30 Agustus 2021

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Pertanyaan yang serupa

Pembahasan Soal:

Cara utama dalam menentukan nilai limit adalah dengan menggunakan metode substitusi. Apabila hasil substitusi tersebut merupakan bentuk tak tentu, selanjutnya penyelesaian harus menggunakan cara lain, yaitu metode pemfaktoran atau kali akar sekawan.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of space fraction numerator square root of 2 x plus 1 end root minus square root of x plus 5 end root over denominator 4 minus x end fraction end cell equals cell fraction numerator square root of 2 times 4 plus 1 end root minus square root of 4 plus 5 end root over denominator 4 minus 4 end fraction end cell row blank equals cell fraction numerator 3 minus 3 over denominator 0 end fraction end cell row blank equals cell 0 over 0 end cell end table

Hasil dari substitusi merupakan bentuk tak tentu sehingga penyelesaian limit tersebut menggunakan perkalian akar sekawan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank 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 4 minus x end fraction end cell row blank 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 4 minus x end fraction cross 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 end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator 2 x plus 1 minus left parenthesis x plus 5 right parenthesis over denominator open parentheses 4 minus x close parentheses open parentheses square root of 2 x plus 1 end root plus square root of x plus 5 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator x minus 4 over denominator open parentheses 4 minus x close parentheses left parenthesis square root of 2 x plus 1 end root plus square root of x plus 5 end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 4 of fraction numerator negative 1 over denominator left parenthesis square root of 2 x plus 1 end root plus square root of x plus 5 end root right parenthesis end fraction end cell row blank equals cell fraction numerator negative 1 over denominator 3 plus 3 end fraction end cell row blank equals cell fraction numerator negative 1 over denominator 6 end fraction end cell end table end style 

Dengan demikian, hasil dari table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of space fraction numerator square root of 2 x plus 1 end root minus square root of x plus 5 end root over denominator 4 minus x end fraction end cell equals cell negative 1 over 6 end cell end table 

Roboguru

....

Pembahasan Soal:

Dengan menggunakan metode substitusi didapatkan

limx24x83x+35x1===886+310109900

Karena menghasilkan nilai 00, maka dapat diselesaikan dengan metode lain, yaitu mengalikan dengan sekawannya sebagai berikut

=======limx24x83x+35x1limx24x83x+35x13x+3+5x13x+3+5x1limx2(4x8)(3x+3+5x1)3x+3(5x1)limx24(x2)(3x+3+5x1)2x+4limx24(x2)(3x+3+5x1)2(x2)limx22(3x+3+5x1)12(3+3)1121

Dengan demikian, nilai dari x2lim4x83x+35x1=121

Roboguru

Nilai  adalah ....

Pembahasan Soal:

Limit ini dapat diselesaikan dengan langsung substitusi sebagai berikut.

limx0x2+5x+1x4====limx0x2+5x+1x4(0)2+5(0)+104144

Dengan demikian, Nilai begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator x minus 4 over denominator x squared plus 5 x plus 1 end fraction end style adalah 4.

Roboguru

....

Pembahasan Soal:

Cara paling dasar dalam menyelesaikan soal limit fungsi aljabar adalah dengan cara substitusi. Apabila hasil dari substitusi tersebut menghasilkan bentuk tak tentu, maka selanjutnya menggunakan cara lain, yaitu dengan faktorisasi dan perkalian sekawan.

Dengan cara substitusi, nilai limit di atas adalah sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space fraction numerator x minus 2 over denominator square root of x squared plus 1 end root minus square root of 5 end fraction end cell equals cell fraction numerator 2 minus 2 over denominator square root of 2 squared plus 1 end root minus square root of 5 end fraction end cell row blank equals cell 0 over 0 end cell end table

Karena hasil substitusi menghasilkan bentuk tak tentu, sehingga nilai limit akan ditentukan dengan cara perkalian akar sekawan.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of space fraction numerator x minus 2 over denominator square root of x squared plus 1 end root minus square root of 5 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator x minus 2 over denominator square root of x squared plus 1 end root minus square root of 5 end fraction cross times fraction numerator square root of x squared plus 1 end root plus square root of 5 over denominator square root of x squared plus 1 end root plus square root of 5 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses x minus 2 close parentheses open parentheses square root of x squared plus 1 end root plus square root of 5 close parentheses over denominator x squared plus 1 minus 5 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses x minus 2 close parentheses open parentheses square root of x squared plus 1 end root plus square root of 5 close parentheses over denominator x squared minus 4 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses x minus 2 close parentheses open parentheses square root of x squared plus 1 end root plus square root of 5 close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses square root of x squared plus 1 end root plus square root of 5 close parentheses over denominator open parentheses x plus 2 close parentheses end fraction end cell row blank equals cell fraction numerator square root of 2 squared plus 1 end root plus square root of 5 over denominator 2 plus 2 end fraction end cell row blank equals cell fraction numerator square root of 5 plus square root of 5 over denominator 4 end fraction end cell row blank equals cell fraction numerator 2 square root of 5 over denominator 4 end fraction end cell row blank equals cell 1 half square root of 5 end cell end table

Dengan demikian, hasil dari table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 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 minus 2 over denominator square root of x squared plus 1 end root minus square root of 5 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 cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 5 end cell end table

Roboguru

Hitunglah nilai setiap limit berikut. b.

Pembahasan Soal:

Perhatikan perhitungan berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 0 of fraction numerator square root of 1 plus x end root minus square root of 1 minus x end root over denominator square root of 2 plus x end root minus square root of 2 minus x end root end fraction end cell equals cell fraction numerator square root of 1 plus 0 end root minus square root of 1 minus 0 end root over denominator square root of 2 plus 0 end root minus square root of 2 minus 0 end root end fraction end cell row blank equals cell fraction numerator square root of 1 minus square root of 1 over denominator square root of 2 minus square root of 2 end fraction end cell row blank equals cell 0 over 0 end cell end table 

Karena hasil limit tersebut ketika disubstitusikan x equals 0 menghasil bilangan tak tentu 0 over 0, maka dilakukan rasionalisasi bentuk akar sekawan sebagai berikut:

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

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

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