Pertanyaan

Nilai dari x → 2 lim x + 7 − 3 1 − ( x − 1 ) 8 adalah ....

Nilai dari $x \to 2 lim \frac{1 - ( x - 1 ) ^{8}}{x + 7 - 3}$ adalah ....

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Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Perhatikan perhitungan berikut! Didapat bentuk .Ingat bahwa merupakan bentuk tak tentu sehingga perlu dicari cara lain untuk mencari nilai limitnya. Perhatikan perhitungan berikut! Dengan demikian, nilai dari adalah . Jadi, jawaban yang tepat adalah E.

Perhatikan perhitungan berikut!

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of invisible function application fraction numerator 1 minus left parenthesis x minus 1 right parenthesis to the power of 8 over denominator square root of x plus 7 end root minus 3 end fraction end cell equals cell fraction numerator limit as x rightwards arrow 2 of open parentheses 1 minus left parenthesis x minus 1 right parenthesis to the power of 8 close parentheses over denominator limit as x rightwards arrow 2 of open parentheses square root of x plus 7 end root minus 3 close parentheses end fraction end cell row blank equals cell fraction numerator 1 minus open parentheses 2 minus 1 close parentheses to the power of 8 over denominator square root of 2 plus 7 end root minus 3 end fraction end cell row blank equals cell 0 over 0 end cell end table end style$

Didapat bentuk . Ingat bahwa merupakan bentuk tak tentu sehingga perlu dicari cara lain untuk mencari nilai limitnya.

Perhatikan perhitungan berikut!

$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 invisible function application fraction numerator 1 minus left parenthesis x minus 1 right parenthesis to the power of 8 over denominator square root of x plus 7 end root minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus open parentheses x minus 1 close parentheses to the power of 4 close parentheses open parentheses 1 minus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses over denominator square root of x plus 7 end root minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses open parentheses 1 minus left parenthesis x minus 1 right parenthesis squared close parentheses over denominator square root of x plus 7 end root minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis close parentheses open parentheses 1 minus left parenthesis x minus 1 right parenthesis close parentheses over denominator square root of x plus 7 end root minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses left parenthesis x right parenthesis left parenthesis 2 minus x right parenthesis over denominator square root of x plus 7 end root minus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application open parentheses fraction numerator open parentheses 1 plus open parentheses x minus 1 close parentheses to the power of 4 close parentheses open parentheses 1 plus open parentheses x minus 1 close parentheses squared close parentheses open parentheses x close parentheses open parentheses 2 minus x close parentheses over denominator square root of x plus 7 end root minus 3 end fraction times fraction numerator square root of x plus 7 end root plus 3 over denominator square root of x plus 7 end root plus 3 end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses left parenthesis x right parenthesis left parenthesis 2 minus x right parenthesis open parentheses square root of x plus 7 end root plus 3 close parentheses over denominator left parenthesis x plus 7 right parenthesis minus 9 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses left parenthesis x right parenthesis left parenthesis 2 minus x right parenthesis open parentheses square root of x plus 7 end root plus 3 close parentheses over denominator x minus 2 end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses left parenthesis x right parenthesis left parenthesis 2 minus x right parenthesis open parentheses square root of x plus 7 end root plus 3 close parentheses over denominator negative left parenthesis 2 minus x right parenthesis end fraction end cell row blank equals cell limit as x rightwards arrow 2 of invisible function application fraction numerator open parentheses 1 plus left parenthesis x minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis x minus 1 right parenthesis squared close parentheses left parenthesis x right parenthesis open parentheses square root of x plus 7 end root plus 3 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus left parenthesis 2 minus 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis 2 minus 1 right parenthesis squared close parentheses left parenthesis 2 right parenthesis open parentheses square root of 2 plus 7 end root plus 3 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus left parenthesis 1 right parenthesis to the power of 4 close parentheses open parentheses 1 plus left parenthesis 1 right parenthesis squared close parentheses left parenthesis 2 right parenthesis open parentheses square root of 9 plus 3 close parentheses over denominator negative 1 end fraction end cell row blank equals cell fraction numerator left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 1 right parenthesis left parenthesis 2 right parenthesis left parenthesis 3 plus 3 right parenthesis over denominator negative 1 end fraction end cell row blank equals cell fraction numerator left parenthesis 2 right parenthesis left parenthesis 2 right parenthesis left parenthesis 2 right parenthesis left parenthesis 6 right parenthesis over denominator negative 1 end fraction end cell row blank equals cell fraction numerator 48 over denominator negative 1 end fraction end cell row blank equals cell negative 48 end cell end table end style$

Dengan demikian, nilai dari $begin mathsize 14px style limit as x rightwards arrow 2 of invisible function application fraction numerator 1 minus left parenthesis x minus 1 right parenthesis to the power of 8 over denominator square root of x plus 7 end root minus 3 end fraction end style$ adalah .

Jadi, jawaban yang tepat adalah E.