Pertanyaan

Hitunglah nilai dari x → 2 lim 2 x + 5 − 5 x − 1 x − 2 .

Hitunglah nilai dari $x \to 2 lim \frac{x - 2}{2 x + 5 - 5 x - 1}$ .

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Jawaban terverifikasi

Jawaban

nilai .

nilai $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 2 over denominator square root of 2 x plus 5 end root minus square root of 5 x minus 1 end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table 2 end style$ .

Pembahasan

Dengan metode substitusi diperoleh sebagai berikut. Berdasarkan uraian di atas, diperoleh hasilnya tak tentu atau , maka nilai limit tersebut harus ditentukan dengan metode yang lain. Dengan metode mengalikan akar sekawan terlebih dahulu lalu substitusi, diperoleh sebagai berikut. Dengan demikian, nilai .

Dengan metode substitusi diperoleh sebagai berikut.

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of fraction numerator x minus 2 over denominator square root of 2 x plus 5 end root minus square root of 5 x minus 1 end root end fraction end cell equals cell fraction numerator 2 minus 2 over denominator square root of 2 times 2 plus 5 end root minus square root of 5 times 2 minus 1 end root end fraction end cell row blank equals cell 0 over 0 end cell end table end style$

Berdasarkan uraian di atas, diperoleh hasilnya tak tentu atau 0 over 0 , maka nilai limit tersebut harus ditentukan dengan metode yang lain. Dengan metode mengalikan akar sekawan terlebih dahulu lalu substitusi, diperoleh sebagai berikut.

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of fraction numerator x minus 2 over denominator square root of 2 x plus 5 end root minus square root of 5 x minus 1 end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator x minus 2 over denominator square root of 2 x plus 5 end root minus square root of 5 x minus 1 end root end fraction times fraction numerator square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root over denominator square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator left parenthesis x minus 2 right parenthesis left parenthesis square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root right parenthesis over denominator left parenthesis 2 x plus 5 right parenthesis minus left parenthesis 5 x minus 1 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator left parenthesis x minus 2 right parenthesis left parenthesis square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root right parenthesis over denominator 2 x plus 5 minus 5 x plus 1 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator left parenthesis x minus 2 right parenthesis left parenthesis square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root right parenthesis over denominator negative 3 x plus 6 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator left parenthesis x minus 2 right parenthesis left parenthesis square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root right parenthesis over denominator negative 3 left parenthesis x minus 2 right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator square root of 2 x plus 5 end root plus square root of 5 x minus 1 end root over denominator negative 3 end fraction end cell row blank equals cell fraction numerator square root of 2 times 2 plus 5 end root plus square root of 5 times 2 minus 1 end root over denominator negative 3 end fraction end cell row blank equals cell fraction numerator 3 plus 3 over denominator negative 3 end fraction end cell row blank equals cell fraction numerator 6 over denominator negative 3 end fraction end cell row blank equals cell negative 2 end cell end table end style$

Dengan demikian, nilai $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator x minus 2 over denominator square root of 2 x plus 5 end root minus square root of 5 x minus 1 end root end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table 2 end style$ .

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