Pertanyaan

Jika x → − 3 lim x 2 − 9 x 2 + a x + b = 2 dan x → 3 lim x 2 − 4 a x 2 + b x + c = 3 . Tentukan nilai .

Jika $x \to - 3 lim \frac{x ^{2} + a x + b}{x ^{2} - 9} = 2$ dan $x \to 3 lim \frac{a x ^{2} + b x + c}{x ^{2} - 4} = 3$ . Tentukan nilai $c$ .

G. Albiah

Master Teacher

Mahasiswa/Alumni Universitas Galuh Ciamis

Jawaban terverifikasi

Jawaban

nilai adalah .

nilai

adalah

Pembahasan

Maka subtitusikan langsung menjadi: Setelah disubtitusikan langsung, dapat diketahui bahwa saat , pada pembilang bernilai bukan bernilai seperti pada soal, sehingga bernilai tak tentu atau . Sehingga, Di dapat persamaan yaitu . Kemudian gunakan aturan L'Hopital dengan menurunkan penyebut dan pembilang untuk mendapatkan nilai limitnya, Berlaku aturan L'Hopital yaitu: Maka, Kemudian subtitusikan ke persamaan , Subtitusikan dan ke , Subtitusikan limit, Jadi,nilai adalah .

$limit as x rightwards arrow negative 3 of space fraction numerator x squared plus a x plus b over denominator x squared minus 9 end fraction equals 2$

Maka subtitusikan langsung menjadi:

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow negative 3 of space fraction numerator x squared plus a x plus b over denominator x squared minus 9 end fraction end cell equals 2 row cell fraction numerator open parentheses 3 close parentheses squared plus a left parenthesis 3 right parenthesis plus b over denominator 3 squared minus 9 end fraction end cell not equal to 2 row cell fraction numerator 9 plus 3 a plus b over denominator 0 end fraction end cell not equal to 2 end table$

Setelah disubtitusikan langsung, dapat diketahui bahwa saat x equals 3 , $fraction numerator x squared plus a x plus b over denominator x squared minus 9 end fraction$ pada pembilang bernilai bukan bernilai seperti pada soal, sehingga $fraction numerator x squared plus a x plus b over denominator x squared minus 9 end fraction$ bernilai tak tentu atau 0 over 0 . Sehingga,

Di dapat persamaan yaitu .

Kemudian gunakan aturan L'Hopital dengan menurunkan penyebut dan pembilang untuk mendapatkan nilai limitnya,

Berlaku aturan L'Hopital yaitu:

$fraction numerator d over denominator d x end fraction x to the power of n equals n times x to the power of n minus 1 end exponent$

Maka,

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow negative 3 of space fraction numerator x squared plus a x plus b over denominator x squared minus 9 end fraction end cell equals 2 row cell limit as x rightwards arrow negative 3 of fraction numerator 2 x to the power of 2 minus 1 end exponent plus a x to the power of 1 minus 1 end exponent plus 0 over denominator 2 x to the power of 2 minus 1 end exponent minus 0 end fraction end cell equals 2 row cell fraction numerator 2 x plus a over denominator 2 x end fraction end cell equals 2 row cell fraction numerator 2 left parenthesis 3 right parenthesis plus a over denominator 2 left parenthesis 3 right parenthesis end fraction end cell equals 2 row cell fraction numerator 6 plus a over denominator 6 end fraction end cell equals 2 row cell 6 plus a end cell equals cell 2 cross times 6 end cell row cell 6 plus a end cell equals 12 row a equals cell 12 minus 6 end cell row a equals 6 row blank blank blank end table$

Kemudian subtitusikan a equals 6 ke persamaan ,

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 a plus b end cell equals cell negative 9 end cell row cell 3 left parenthesis 6 right parenthesis plus b end cell equals cell negative 9 end cell row cell 18 plus b end cell equals cell negative 9 end cell row b equals cell negative 9 minus 18 end cell row b equals cell negative 27 end cell end table

Subtitusikan a equals 6 dan b equals negative 27 ke $limit as x rightwards arrow 3 of fraction numerator a x squared plus b x plus c over denominator x squared minus 4 end fraction equals 3$ ,

$limit as x rightwards arrow 3 of fraction numerator a x squared plus b x plus c over denominator x squared minus 4 end fraction equals 3 limit as x rightwards arrow 3 of fraction numerator 6 x squared minus 27 x plus c over denominator x squared minus 4 end fraction equals 3$

Subtitusikan limit,

$table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 3 of fraction numerator 6 x squared minus 27 x plus c over denominator x squared minus 4 end fraction end cell equals 3 row cell limit as x rightwards arrow 3 of fraction numerator 6 left parenthesis 3 right parenthesis squared minus 27 left parenthesis 3 right parenthesis plus c over denominator left parenthesis 3 right parenthesis squared minus 4 end fraction end cell equals 3 row cell fraction numerator 6 left parenthesis 9 right parenthesis minus 81 plus c over denominator 9 minus 4 end fraction end cell equals 3 row cell fraction numerator 54 minus 81 plus c over denominator 5 end fraction end cell equals 3 row cell 54 minus 81 plus c end cell equals cell 3 cross times 5 end cell row cell negative 27 plus c end cell equals 15 row c equals cell 15 plus 27 end cell row c equals 42 row blank blank blank end table$

Jadi, nilai adalah .