Pertanyaan

Tentukan limit fungsi berikut: x → 1 lim x − 1 x − 2 x − 1

Tentukan limit fungsi berikut:

$x \to 1 lim \frac{x - 2 x - 1}{x - 1}$

R. Hajrianti

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

nilai limit fungsi adalah .

nilai limit fungsi $limit as x rightwards arrow 1 of fraction numerator square root of x minus square root of 2 x minus 1 end root over denominator x minus 1 end fraction$ adalah negative 1 half

Pembahasan

Langkah pertama untuk menentukan nilai limit fungsi adalah mengalikan dengan sekawan sebagai berikut. Kemudian, dengan menggunakan metode substitusi, diperoleh nilai limit tersebut sebagai berikut. Dengan demikian, nilai limit fungsi adalah .

Langkah pertama untuk menentukan nilai limit fungsi $limit as x rightwards arrow 1 of fraction numerator square root of x minus square root of 2 x minus 1 end root over denominator x minus 1 end fraction$ adalah mengalikan dengan sekawan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 1 of fraction numerator square root of x minus square root of 2 x minus 1 end root over denominator x minus 1 end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator square root of x minus square root of 2 x minus 1 end root over denominator x minus 1 end fraction times fraction numerator open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses over denominator open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator open parentheses square root of x close parentheses squared minus open parentheses square root of 2 x minus 1 end root close parentheses squared over denominator open parentheses x minus 1 close parentheses open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator x minus open parentheses 2 x minus 1 close parentheses over denominator open parentheses x minus 1 close parentheses open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator x minus 2 x plus 1 over denominator open parentheses x minus 1 close parentheses open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative x plus 1 over denominator open parentheses x minus 1 close parentheses open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative up diagonal strike open parentheses x minus 1 close parentheses end strike to the power of 1 over denominator up diagonal strike open parentheses x minus 1 close parentheses subscript 1 end strike open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 1 of fraction numerator negative 1 over denominator open parentheses square root of x plus square root of 2 x minus 1 end root close parentheses end fraction end cell end table$

Kemudian, dengan menggunakan metode substitusi, diperoleh nilai limit tersebut sebagai berikut.

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

Dengan demikian, nilai limit fungsi $limit as x rightwards arrow 1 of fraction numerator square root of x minus square root of 2 x minus 1 end root over denominator x minus 1 end fraction$ adalah negative 1 half .