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Tentukan x→1lim​x−13−x+8​​!

Pertanyaan

Tentukan limit as x rightwards arrow 1 of fraction numerator 3 minus square root of x plus 8 end root over denominator x minus 1 end fraction!

H. Endah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Jawaban

begin mathsize 14px style limit as straight x rightwards arrow 1 of fraction numerator 3 minus square root of straight x plus 8 end root over denominator straight x minus 1 end fraction equals negative 1 over 6 end style.

Pembahasan

Perhatikan perhitungan berikut!

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 1 of fraction numerator 3 minus square root of x plus 8 end root over denominator x minus 1 end fraction end cell equals cell fraction numerator 3 minus square root of open parentheses 1 close parentheses plus 8 end root over denominator open parentheses 1 close parentheses minus 1 end fraction end cell row blank equals cell fraction numerator 3 minus square root of 9 over denominator 1 minus 1 end fraction end cell row blank equals cell fraction numerator 3 minus 3 over denominator 1 minus 1 end fraction end cell row blank equals cell 0 over 0 end cell end table 

Karena hasil dari limit tersebut ketika disubstitusikan x equals 1 merupakan bilangan tak tentu 0 over 0, maka kita lakukan dengan mengalikan akar sekawan pada bentuk limit tersebut.

Hasil perhitungan dengan mengalikan akar sekawan dari pembilangnya adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as straight x rightwards arrow 1 of fraction numerator 3 minus square root of straight x plus 8 end root over denominator straight x minus 1 end fraction times fraction numerator 3 plus square root of straight x plus 8 end root over denominator 3 plus square root of straight x plus 8 end root end fraction end cell equals cell limit as straight x rightwards arrow 1 of fraction numerator 9 minus open parentheses straight x plus 8 close parentheses over denominator open parentheses straight x minus 1 close parentheses open parentheses 3 plus square root of straight x plus 8 end root close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow 1 of fraction numerator 9 minus x minus 8 over denominator open parentheses straight x minus 1 close parentheses open parentheses 3 plus square root of straight x plus 8 end root close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow 1 of fraction numerator negative x plus 1 over denominator open parentheses straight x minus 1 close parentheses open parentheses 3 plus square root of straight x plus 8 end root close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow 1 of fraction numerator negative up diagonal strike open parentheses straight x minus 1 close parentheses end strike over denominator up diagonal strike open parentheses straight x minus 1 close parentheses end strike open parentheses 3 plus square root of straight x plus 8 end root close parentheses end fraction end cell row blank equals cell limit as straight x rightwards arrow 1 of fraction numerator negative 1 over denominator open parentheses 3 plus square root of straight x plus 8 end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 1 over denominator 3 plus square root of 1 plus 8 end root end fraction end cell row blank equals cell fraction numerator negative 1 over denominator 3 plus square root of 9 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 negative 1 over 6 end cell end table 

Dengan demikian, begin mathsize 14px style limit as straight x rightwards arrow 1 of fraction numerator 3 minus square root of straight x plus 8 end root over denominator straight x minus 1 end fraction equals negative 1 over 6 end style.

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