Tentukan nilai x→2lim​ (x−2x−8−2x​​).

Pertanyaan

Tentukan nilai limit as x rightwards arrow 2 of space open parentheses fraction numerator x minus square root of 8 minus 2 x end root over denominator x minus 2 end fraction close parentheses.

T. Prita

Master Teacher

Mahasiswa/Alumni Universitas Negeri Jember

Jawaban terverifikasi

Jawaban

 nilai limit as x rightwards arrow 2 of space open parentheses fraction numerator x minus square root of 8 minus 2 x end root over denominator x minus 2 end fraction close parentheses adalah 1 comma 5.

Pembahasan

Dalam mengerjakan limit, jika substitusi langsung menghasilkan bentuk tak tentu maka harus menggunakan cara lain dalam mencari nilainya menggunakan operasi aljabar misalnya memfaktorkan, membagi, merasionalkan, dan lainnya.

Beberapa bentuk tak tentu antara lain: 0 over 0 comma space infinity over infinity comma space open parentheses infinity minus infinity close parentheses comma space open parentheses 0 times infinity close parentheses

Substitusi nilai x equals 2 diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space open parentheses fraction numerator x minus square root of 8 minus 2 x end root over denominator x minus 2 end fraction close parentheses end cell equals cell fraction numerator 2 minus square root of 8 minus 2 open parentheses 2 close parentheses end root over denominator 2 minus 2 end fraction end cell row blank equals cell fraction numerator 2 minus square root of 4 over denominator 2 minus 2 end fraction end cell row blank equals cell fraction numerator 2 minus 2 over denominator 2 minus 2 end fraction end cell row blank equals cell 0 over 0 space open parentheses text bentuk tak tentu end text close parentheses end cell end table

Selanjutnya dengan mengkalikan akar sekawan diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of space open parentheses fraction numerator x minus square root of 8 minus 2 x end root over denominator x minus 2 end fraction close parentheses end cell equals cell limit as x rightwards arrow 2 of space open parentheses fraction numerator x minus square root of 8 minus 2 x end root over denominator x minus 2 end fraction close parentheses times fraction numerator x plus square root of 8 minus 2 x end root over denominator x plus square root of 8 minus 2 x end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of open parentheses fraction numerator open parentheses x minus square root of 8 minus 2 x end root close parentheses open parentheses x plus square root of 8 minus 2 x end root close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses x plus square root of 8 minus 2 x end root close parentheses end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of open parentheses fraction numerator x squared minus open parentheses 8 minus 2 x close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses x plus square root of 8 minus 2 x end root close parentheses end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of open parentheses fraction numerator x squared minus 8 plus 2 x over denominator open parentheses x minus 2 close parentheses open parentheses x plus square root of 8 minus 2 x end root close parentheses end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of open parentheses fraction numerator x squared plus 2 x minus 8 over denominator open parentheses x minus 2 close parentheses open parentheses x plus square root of 8 minus 2 x end root close parentheses end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of open parentheses fraction numerator open parentheses x plus 4 close parentheses open parentheses x minus 2 close parentheses over denominator open parentheses x minus 2 close parentheses open parentheses x plus square root of 8 minus 2 x end root close parentheses end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of space fraction numerator open parentheses x plus 4 close parentheses over denominator open parentheses x plus square root of 8 minus 2 x end root close parentheses end fraction end cell row blank equals cell fraction numerator open parentheses 2 plus 4 close parentheses over denominator open parentheses 2 plus square root of 8 minus 2 open parentheses 2 close parentheses end root close parentheses end fraction end cell row blank equals cell fraction numerator 6 over denominator 2 plus square root of 4 end fraction end cell row blank equals cell fraction numerator 6 over denominator 2 plus 2 end fraction end cell row blank equals cell 6 over 4 end cell row blank equals cell 3 over 2 end cell row blank equals cell 1 comma 5 end cell end table

Dengan demikian nilai limit as x rightwards arrow 2 of space open parentheses fraction numerator x minus square root of 8 minus 2 x end root over denominator x minus 2 end fraction close parentheses adalah 1 comma 5.

94

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