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PERTANYAAN :

begin mathsize 14px style limit as x rightwards arrow 0 of fraction numerator x over denominator square root of 1 plus x end root minus square root of 1 minus x end root end fraction equals end style….

 

Jawaban

nilai dari limit as x rightwards arrow 0 of fraction numerator x over denominator square root of 1 plus x end root minus square root of 1 minus x end root end fraction equals 1.


Pembahasan

Perhatikan bahwa:

Saat dilakukan substitusi, limit di atas akan bernilai

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

Karena hasilnya masih dalam bentuk 0 over 0, maka selesaikan dengan metode mengalikan akar sekawan.

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

Dengan demikian, nilai dari limit as x rightwards arrow 0 of fraction numerator x over denominator square root of 1 plus x end root minus square root of 1 minus x end root end fraction equals 1.


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