Roboguru

Nilai

Pertanyaan

Nilai limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction equals....

  1. 1 fourth space

  2. 1 half space

  3. 3 over 4 space

  4. 2 space

  5. 4 space

Pembahasan Soal:

Nilai limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction dapat ditentukan dengan mensubstitusi nilai x equals 1 ke dalam fungsi fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction sehingga

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

Karena hasil limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction menggunakan konsep limit yaitu cara substitusi adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 0 over 0 comma end cell end table maka untuk menentukan limitnya menggunakan cara selanjutnya yaitu mengalikan dengan 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 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction cross times fraction numerator square root of 5 x minus 4 end root plus x over denominator square root of 5 x minus 4 end root plus x end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator open parentheses 5 x minus 4 close parentheses plus up diagonal strike x square root of 5 x minus 4 end root minus x square root of 5 x minus 4 end root end strike minus x squared over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses open parentheses square root of 5 x minus 4 end root plus x close parentheses end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator 5 x minus 4 minus x squared over denominator open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses open parentheses square root of 5 x minus 4 end root plus x close parentheses end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator up diagonal strike open parentheses x minus 1 close parentheses end strike left parenthesis negative x plus 4 right parenthesis over denominator up diagonal strike open parentheses x minus 1 close parentheses end strike open parentheses x plus 1 close parentheses open parentheses square root of 5 x minus 4 end root plus x close parentheses end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator negative x plus 4 over denominator open parentheses x plus 1 close parentheses open parentheses square root of 5 x minus 4 end root plus x close parentheses end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell fraction numerator negative 1 plus 4 over denominator open parentheses 1 plus 1 close parentheses open parentheses square root of 5 open parentheses 1 close parentheses minus 4 end root plus 1 close parentheses end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell fraction numerator 3 over denominator 2 open parentheses square root of 1 plus 1 close parentheses end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell fraction numerator 3 over denominator 2 cross times 2 end fraction end cell row cell limit as x rightwards arrow 1 of fraction numerator square root of 5 x minus 4 end root minus x over denominator x squared minus 1 end fraction end cell equals cell 3 over 4 end cell end table end style


Oleh karena itu, jawaban yang benar adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. Febrianti

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 06 Juni 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Tentukan nilai limit berikut.

Pembahasan Soal:

Mencari nilai limit:

limit as x rightwards arrow thin space 0 of open parentheses fraction numerator square root of 8 plus x end root minus square root of 8 minus 3 x end root over denominator 2 x end fraction close parentheses equals 1 half times limit as x rightwards arrow thin space 0 of open parentheses fraction numerator square root of 8 plus x end root minus square root of 8 minus 3 x end root over denominator x end fraction close parentheses equals 1 half times limit as x rightwards arrow thin space 0 of open parentheses fraction numerator square root of 8 plus x end root minus square root of 8 minus 3 x end root cross times fraction numerator square root of 8 plus x end root plus square root of 8 minus 3 x end root over denominator square root of 8 plus x end root plus square root of 8 minus 3 x end root end fraction over denominator x end fraction close parentheses equals 1 half times limit as x rightwards arrow thin space 0 of open parentheses fraction numerator fraction numerator 8 plus x minus 8 plus 3 x over denominator square root of 8 plus x end root plus square root of 8 minus 3 x end root end fraction over denominator x end fraction close parentheses equals 1 half times limit as x rightwards arrow thin space 0 of open parentheses fraction numerator fraction numerator 4 x over denominator square root of 8 plus x end root plus square root of 8 minus 3 x end root end fraction over denominator x end fraction close parentheses equals 1 half times limit as x rightwards arrow thin space 0 of open parentheses fraction numerator 4 over denominator square root of x plus 8 end root plus square root of negative 3 x plus 8 end root end fraction close parentheses equals 1 half times fraction numerator 4 over denominator square root of 0 plus 8 end root plus square root of negative 3 times thin space 0 plus 8 end root end fraction equals 1 half times fraction numerator 4 over denominator square root of 8 plus square root of 8 end fraction equals 1 half times fraction numerator 4 over denominator 2 square root of 2 plus 2 square root of 2 end fraction equals 1 half times fraction numerator 4 over denominator 4 square root of 2 end fraction equals 1 half times fraction numerator 1 over denominator square root of 2 end fraction equals fraction numerator 1 over denominator 2 square root of 2 end fraction 

Jadi, limit as x rightwards arrow thin space 0 of open parentheses fraction numerator square root of 8 plus x end root minus square root of 8 minus 3 x end root over denominator 2 x end fraction close parentheses equals fraction numerator 1 over denominator 2 square root of 2 end fraction.

0

Roboguru

Nilai  adalah ....

Pembahasan Soal:

Limit tersebut jika dimasukkan nilai begin mathsize 14px style x end style mendekati begin mathsize 14px style 1 end style akan diperoleh bentuk tak tentu begin mathsize 14px style 0 over 0 end style, sehingga harus dikalikan dengan akar sekawan kemudian disederhakanakan.

Perhatikan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell limit as x rightwards arrow 1 of fraction numerator 9 square root of x minus 9 over denominator x minus 1 end fraction end cell equals cell limit as x rightwards arrow 1 of fraction numerator 9 square root of x minus 9 over denominator x minus 1 end fraction cross times fraction numerator 9 square root of x plus 9 over denominator 9 square root of x plus 9 end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 81 x minus 81 over denominator open parentheses x minus 1 close parentheses open parentheses 9 square root of x plus 9 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 81 open parentheses x minus 1 close parentheses over denominator open parentheses x minus 1 close parentheses open parentheses 9 square root of x plus 9 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator 81 over denominator open parentheses 9 square root of x plus 9 close parentheses end fraction end cell row blank equals cell fraction numerator 81 over denominator 9 square root of 1 plus 9 end fraction end cell row blank equals cell 81 over 18 end cell row cell limit as x rightwards arrow 1 of fraction numerator 9 square root of x minus 9 over denominator x minus 1 end fraction end cell equals cell 9 over 2 end cell end table end style  

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Pembahasan Soal:

Limit begin mathsize 14px style f open parentheses x close parentheses end style mendekati undefined sama dengan undefined, ditulis:

begin mathsize 14px style limit as x rightwards arrow c of space f open parentheses x close parentheses equals L end style  

jika untuk setiap undefined yang cukup dekat dengan undefined tetapi begin mathsize 14px style x not equal to c end styleundefined mendekati undefined.

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

0

Roboguru

Pembahasan Soal:

Untuk mengetahui nilai dari limit fungsi aljabar yaitu dengan substitusi nilai x ke dalam fungsi tersebut sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator x squared minus 16 over denominator square root of x minus 4 end root end fraction end cell equals cell fraction numerator 4 squared minus 16 over denominator square root of 4 minus 4 end root end fraction end cell row blank equals cell fraction numerator 16 minus 16 over denominator square root of 0 end fraction end cell row blank equals cell fraction numerator 0 over denominator square root of 0 end fraction end cell row blank equals cell 0 over 0 end cell row blank equals cell tak space tentu end cell end table

Karena hasil dari substitusi yaitu tak tentu, maka harus menggunakan cara yang lain untuk menentukan nilai dari limit fungsi tersebut.

Ingat jika terdapat limit fungsi aljabar dengan bentuk rasional dengan penyebutnya bentuk akar maka dapat dilakukan perkalian sekawan terlebih dahulu sebelum melakukan substitusi dan ingat juga dengan menggunakan metode pemfaktoran sehinnga akan didapatkan hasil sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow 4 of fraction numerator straight x squared minus 16 over denominator square root of straight x minus 4 end root end fraction end cell equals cell limit as straight x rightwards arrow 4 of fraction numerator straight x squared minus 16 over denominator square root of straight x minus 4 end root end fraction cross times fraction numerator square root of straight x minus 4 end root over denominator square root of straight x minus 4 end root end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator open parentheses straight x squared minus 16 close parentheses open parentheses square root of straight x minus 4 end root close parentheses over denominator square root of straight x minus 4 end root cross times square root of straight x minus 4 end root end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator open parentheses straight x squared minus 16 close parentheses open parentheses square root of straight x minus 4 end root close parentheses over denominator straight x minus 4 end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator left parenthesis straight x plus 4 right parenthesis left parenthesis x minus 4 right parenthesis left parenthesis square root of straight x minus 4 end root right parenthesis over denominator left parenthesis x minus 4 right parenthesis end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of fraction numerator left parenthesis straight x plus 4 right parenthesis up diagonal strike open parentheses straight x minus 4 close parentheses end strike left parenthesis square root of straight x minus 4 end root right parenthesis over denominator up diagonal strike straight x minus 4 end strike end fraction end cell row blank equals cell limit as straight x rightwards arrow 4 of left parenthesis straight x plus 4 right parenthesis left parenthesis square root of straight x minus 4 end root right parenthesis end cell row blank equals cell left parenthesis 4 plus 4 right parenthesis left parenthesis square root of 4 minus 4 end root right parenthesis end cell row blank equals cell 8 cross times square root of 0 end cell row blank equals cell 8 cross times 0 end cell row blank equals 0 end table

Jadi, dapat disimpulkan bahwa hasil dari  limit as x rightwards arrow 4 of fraction numerator x squared minus 16 over denominator square root of x minus 4 end root end fraction adalah 0.

0

Roboguru

Hitunglah

Pembahasan Soal:

begin mathsize 14px style 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 x 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 equals limit as x rightwards arrow 0 of fraction numerator open parentheses 1 plus x close parentheses minus open parentheses 1 minus x close parentheses over denominator x open parentheses square root of 1 plus x end root plus square root of 1 minus x end root close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator 2 x over denominator x open parentheses square root of 1 plus x end root plus square root of 1 minus x end root close parentheses end fraction equals limit as x rightwards arrow 0 of fraction numerator 2 over denominator square root of 1 plus x end root plus square root of 1 minus x end root end fraction equals fraction numerator 2 over denominator square root of 1 plus 0 end root plus square root of 1 minus 0 end root end fraction equals 2 over 2 equals 1 end style

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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