Nilai dari x→0lim9+x−9−x3x adalah ....

Pertanyaan

Nilai dari $begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator 3 x over denominator square root of 9 plus x end root minus square root of 9 minus x end root end fraction end style$ adalah ....

H. Endah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

nilai dari $begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator 3 x over denominator square root of 9 plus x end root minus square root of 9 minus x end root end fraction end style$ adalah $9$ .

Pembahasan

Perhatikan perhitungan berikut!

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

Karena hasil dari limit tersebut ketika disubstitusikan $x=0$ merupakan bilangan tak tentu 0 over 0 , maka kita lakukan mengalikan akar sekawan pada bentuk limit tersebut.

Sehingga diperoleh hasil perhitungan 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 3 x over denominator square root of 9 plus x end root minus square root of 9 minus x end root end fraction end cell equals cell limit as x rightwards arrow 0 of fraction numerator 3 x over denominator square root of 9 plus x end root minus square root of 9 minus x end root end fraction cross times fraction numerator square root of 9 plus x end root plus square root of 9 minus x end root over denominator square root of 9 plus x end root plus square root of 9 minus x end root end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 x open parentheses square root of 9 plus x end root plus square root of 9 minus x end root close parentheses over denominator open parentheses square root of 9 plus x end root minus square root of 9 minus x end root close parentheses open parentheses square root of 9 plus x end root plus square root of 9 minus x end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 x open parentheses square root of 9 plus x end root plus square root of 9 minus x end root close parentheses over denominator open parentheses 9 plus x close parentheses minus open parentheses 9 minus x close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 x open parentheses square root of 9 plus x end root plus square root of 9 minus x end root close parentheses over denominator 9 plus x minus 9 plus x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 up diagonal strike x open parentheses square root of 9 plus x end root plus square root of 9 minus x end root close parentheses over denominator 2 up diagonal strike x end fraction end cell row blank equals cell limit as x rightwards arrow 0 of fraction numerator 3 open parentheses square root of 9 plus x end root plus square root of 9 minus x end root close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 open parentheses square root of 9 plus open parentheses 0 close parentheses end root plus square root of 9 minus open parentheses 0 close parentheses end root close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 open parentheses square root of 9 plus square root of 9 close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 open parentheses 3 plus 3 close parentheses over denominator 2 end fraction end cell row blank equals cell fraction numerator 3 open parentheses 6 close parentheses over denominator 2 end fraction end cell row blank equals cell 18 over 2 end cell row blank equals 9 end table end style$

Dengan demikian, nilai dari $begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator 3 x over denominator square root of 9 plus x end root minus square root of 9 minus x end root end fraction end style$ adalah $9$ .

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