Pertanyaan

Tentukan nilai dari x → 0 lim x 2 + x − 2 − x

Tentukan nilai dari $x \to 0 lim \frac{2 + x - 2 - x}{x}$

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Klaim

J. Joko

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Jawaban terverifikasi

Jawaban

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

Pembahasan

Substitusi Karena didapat bentuk yang merupakan bentuk tak tentu, maka dikali akar sekawan dahulu. Jadi, .

Substitusi

Karena didapat bentuk yang merupakan bentuk tak tentu, maka dikali akar sekawan dahulu.

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

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