Jika f(x)=a→0limax+a1−x1 dan f(b)=6−1 maka b=...

Pertanyaan

Jika $f open parentheses x close parentheses equals limit as a rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator x plus a end fraction end style minus begin display style 1 over x end style over denominator a end fraction$ dan $f open parentheses b close parentheses equals fraction numerator negative 1 over denominator 6 end fraction$ maka $b=...$

$fraction numerator 1 over denominator square root of 6 end fraction$ atau $fraction numerator negative 1 over denominator square root of 6 end fraction$
$\frac16$ atau $fraction numerator negative 1 over denominator 6 end fraction$
$\sqrt6$ atau $-\sqrt6$
$3$
$6$

Y. Herlanda

Master Teacher

Mahasiswa/Alumni STKIP PGRI Jombang

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah C.

Menentukan Nilai Limit Fungsi Aljabar dengan Metode Pemfaktoran

$f open parentheses x close parentheses equals limit as a rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator x plus a end fraction end style minus begin display style 1 over x end style over denominator a end fraction$

Substitusikan nilai $a=0$ ke fungsi di atas, sehingga diperoleh:

Karena $\frac00$ merupakan bentuk tak tentu, maka bentuk tersebut disederhanakan menjadi:

$table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell limit as a rightwards arrow 0 of fraction numerator begin display style fraction numerator 1 over denominator x plus a end fraction end style minus begin display style 1 over x end style over denominator a end fraction end cell row blank equals cell limit as a rightwards arrow 0 of fraction numerator begin display style fraction numerator x minus x plus a over denominator x open parentheses x plus a close parentheses end fraction end style over denominator a end fraction end cell row blank equals cell limit as a rightwards arrow 0 of fraction numerator begin display style fraction numerator a over denominator x open parentheses x plus a close parentheses end fraction end style over denominator a end fraction end cell row blank equals cell limit as a rightwards arrow 0 of fraction numerator up diagonal strike a over denominator x open parentheses x plus a close parentheses end fraction cross times fraction numerator 1 over denominator up diagonal strike a end fraction end cell row blank equals cell limit as a rightwards arrow 0 of fraction numerator 1 over denominator x open parentheses x plus a close parentheses end fraction end cell end table$

Jika $f open parentheses b close parentheses equals fraction numerator negative 1 over denominator 6 end fraction$ maka $b=...$

$table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses b close parentheses end cell equals cell limit as a rightwards arrow 0 of fraction numerator 1 over denominator x open parentheses x plus a close parentheses end fraction end cell row cell fraction numerator negative 1 over denominator 6 end fraction end cell equals cell fraction numerator 1 over denominator x open parentheses x plus 0 close parentheses end fraction end cell row cell fraction numerator negative 1 over denominator 6 end fraction end cell equals cell 1 over x squared end cell row cell negative x squared end cell equals 6 row cell x squared end cell equals cell negative 6 end cell row cell x squared end cell equals cell 6 i end cell row x equals cell plus-or-minus square root of 6 i end root end cell row cell x subscript 1 end cell equals cell square root of 6 i end root end cell row blank blank atau row cell x subscript 2 end cell equals cell negative square root of 6 i end root end cell end table$

Penyelesaian dalam bentuk imajiner, sehingga tidak ada pilihan jawaban yang tepat.

Jika $f\left(b\right)=\frac16$ , maka:

$table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses b close parentheses end cell equals cell limit as a rightwards arrow 0 of fraction numerator 1 over denominator x open parentheses x plus a close parentheses end fraction end cell row cell 1 over 6 end cell equals cell fraction numerator 1 over denominator x open parentheses x plus 0 close parentheses end fraction end cell row cell 1 over 6 end cell equals cell 1 over x squared end cell row cell x squared end cell equals 6 row x equals cell plus-or-minus square root of 6 end cell row cell x subscript 1 end cell equals cell square root of 6 end cell row blank blank atau row cell x subscript 2 end cell equals cell negative square root of 6 end cell end table$

Dengan demikian, diperoleh nilai $b$ yaitu $\sqrt6$ atau $-\sqrt6$ .

Oleh karena itu, jawaban yang benar adalah C.