Pertanyaan

x → 3 lim x − 3 x + 4 − 2 x + 1 = . . .

$x \to 3 lim \frac{x + 4 - 2 x + 1}{x - 3} = . . .$

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Jawaban terverifikasi

Jawaban

nilai limit tersebut adalah .

nilai limit tersebut adalah $begin mathsize 14px style negative fraction numerator square root of 7 over denominator 14 end fraction end style$ .

Pembahasan

Dengan menggunakan metode substitusi, kita peroleh limit fungsi yang diberikan berbentuk tak tentu sebagai berikut. Oleh karena itu, kitamenggunakan metode mengalikan akar sekawanuntuk memperoleh limit fungsiyang diberikansebagai berikut. Jadi, nilai limit tersebut adalah .

Dengan menggunakan metode substitusi, kita peroleh limit fungsi yang diberikan berbentuk tak tentu sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 3 of fraction numerator square root of x plus 4 end root minus square root of 2 x plus 1 end root over denominator x minus 3 end fraction end cell equals cell fraction numerator square root of 3 plus 4 end root minus square root of 2 times 3 plus 1 end root over denominator 3 minus 3 end fraction end cell row blank equals cell fraction numerator square root of 7 minus square root of 7 over denominator 3 minus 3 end fraction end cell row blank equals cell 0 over 0 end cell end table end style$

Oleh karena itu, kita menggunakan metode mengalikan akar sekawan untuk memperoleh limit fungsi yang diberikan sebagai berikut.

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 3 of fraction numerator square root of x plus 4 end root minus square root of 2 x plus 1 end root over denominator x minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator left parenthesis square root of x plus 4 end root minus square root of 2 x plus 1 end root right parenthesis over denominator x minus 3 end fraction times fraction numerator left parenthesis square root of x plus 4 end root plus square root of 2 x plus 1 end root right parenthesis over denominator left parenthesis square root of x plus 4 end root plus square root of 2 x plus 1 end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator left parenthesis square root of x plus 4 end root right parenthesis squared minus left parenthesis square root of 2 x plus 1 end root right parenthesis squared over denominator left parenthesis x minus 3 right parenthesis left parenthesis square root of x plus 4 end root plus square root of 2 x plus 1 end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator x plus 4 minus left parenthesis 2 x plus 1 right parenthesis over denominator left parenthesis x minus 3 right parenthesis left parenthesis square root of x plus 4 end root plus square root of 2 x plus 1 end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator negative x plus 3 over denominator left parenthesis x minus 3 right parenthesis left parenthesis square root of x plus 4 end root plus square root of 2 x plus 1 end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator negative left parenthesis x minus 3 right parenthesis over denominator left parenthesis x minus 3 right parenthesis left parenthesis square root of x plus 4 end root plus square root of 2 x plus 1 end root right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator negative 1 over denominator square root of x plus 4 end root plus square root of 2 x plus 1 end root end fraction end cell row blank equals cell fraction numerator negative 1 over denominator square root of 3 plus 4 end root plus square root of 2 times 3 plus 1 end root end fraction end cell row blank equals cell fraction numerator negative 1 over denominator square root of 7 plus square root of 7 end fraction end cell row blank equals cell negative fraction numerator 1 over denominator 2 square root of 7 end fraction equals negative fraction numerator 1 over denominator 2 square root of 7 end fraction times fraction numerator square root of 7 over denominator square root of 7 end fraction end cell row blank equals cell negative fraction numerator square root of 7 over denominator 14 end fraction end cell end table end style$

Jadi, nilai limit tersebut adalah $begin mathsize 14px style negative fraction numerator square root of 7 over denominator 14 end fraction end style$ .