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Misalkan, x→4lim​x2−16ax2+bx−x​​=21​. Maka bilangan bulat terbesar yang lebih kecil atau sama dengan a−2b adalah...

Pertanyaan

Misalkan, limit as x rightwards arrow 4 of fraction numerator a x squared plus b x minus square root of x over denominator x squared minus 16 end fraction equals 1 half. Maka bilangan bulat terbesar yang lebih kecil atau sama dengan a minus 2 b adalah...

  1. negative 5 

  2. 2 

  3. 6 

  4.   7 

  5. 8 

Y. Herlanda

Master Teacher

Mahasiswa/Alumni STKIP PGRI Jombang

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah D.

Menentukan Nilai Limit Fungsi Aljabar dengan Aturan L'Hospital

Substitusikan x equals 4, ke dalam fungsi berikut. Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator a x squared plus b x minus square root of x over denominator x squared minus 16 end fraction end cell equals cell 1 half end cell row cell fraction numerator a open parentheses 4 close parentheses squared plus b open parentheses 4 close parentheses minus square root of 4 over denominator open parentheses 4 close parentheses squared minus 16 end fraction end cell equals cell 1 half end cell row cell fraction numerator 16 a plus 4 b minus 2 over denominator 16 minus 16 end fraction end cell equals cell 1 half end cell row cell 2 open parentheses 16 a plus 4 b minus 2 close parentheses end cell equals 0 row cell 32 a plus 8 b minus 4 end cell equals 0 row cell 8 a plus 2 b minus 1 end cell equals 0 row cell 8 a plus 2 b end cell equals 1 end table 

Dengan menggunakan aturan L'Hospital diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 4 of fraction numerator a x squared plus b x minus square root of x over denominator x squared minus 16 end fraction end cell equals cell 1 half end cell row cell limit as x rightwards arrow 4 of fraction numerator a x squared plus b x minus open parentheses x close parentheses to the power of begin display style 1 half end style end exponent over denominator x squared minus 16 end fraction end cell equals cell 1 half end cell row cell limit as x rightwards arrow 4 of fraction numerator 2 a x plus b minus begin display style 1 half end style x to the power of negative begin display style 1 half end style end exponent over denominator 2 x end fraction end cell equals cell 1 half end cell row cell limit as x rightwards arrow 4 of fraction numerator 2 a x plus b minus begin display style fraction numerator 1 over denominator 2 square root of x end fraction end style over denominator 2 x end fraction end cell equals cell 1 half end cell row cell fraction numerator 2 a open parentheses 4 close parentheses plus b minus begin display style fraction numerator 1 over denominator 2 square root of 4 end fraction end style over denominator 2 open parentheses 4 close parentheses end fraction end cell equals cell 1 half end cell row cell fraction numerator 8 a plus b minus begin display style fraction numerator 1 over denominator 2 open parentheses 2 close parentheses end fraction end style over denominator 8 end fraction end cell equals cell 1 half end cell row cell 2 open parentheses 8 a plus b minus 1 fourth close parentheses end cell equals 8 row cell 16 a plus 2 b minus 1 half end cell equals 8 row cell 32 a plus 4 b minus 1 end cell equals 16 row cell 32 a plus 4 b end cell equals 17 row cell 8 a plus b end cell equals cell 4 1 fourth end cell row b equals cell 4 1 fourth minus 8 a end cell end table 

Substitusi b equals 4 1 fourth minus 8 a ke 8 a plus 2 b equals 1:

table attributes columnalign right center left columnspacing 0px end attributes row cell 8 a plus 2 b end cell equals 1 row cell 8 a plus 2 open parentheses 4 1 fourth minus 8 a close parentheses end cell equals 1 row cell 8 a plus 2 open parentheses 17 over 4 minus 8 a close parentheses end cell equals 1 row cell 8 a plus 34 over 4 minus 16 a end cell equals 1 row cell negative 8 a end cell equals cell 1 minus 34 over 4 end cell row cell negative 8 a end cell equals cell 4 over 4 minus 34 over 4 end cell row cell negative 8 a end cell equals cell negative 30 over 4 end cell row a equals cell negative 30 over 4 cross times open parentheses negative 1 over 8 close parentheses end cell row blank equals cell 30 over 32 end cell row blank equals cell 15 over 16 end cell end table 

Karena a equals 15 over 16, maka:

table attributes columnalign right center left columnspacing 0px end attributes row b equals cell 4 1 fourth minus 8 a end cell row blank equals cell 17 over 4 minus 8 open parentheses 15 over 16 close parentheses end cell row blank equals cell 17 over 4 minus 120 over 16 end cell row blank equals cell fraction numerator 68 minus 120 over denominator 16 end fraction end cell row blank equals cell fraction numerator negative 52 over denominator 16 end fraction end cell row blank equals cell negative 13 over 4 end cell end table 

Karena a equals 15 over 16b equals negative 13 over 4 maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell a minus 2 b end cell equals cell 15 over 16 minus 2 open parentheses negative 13 over 4 close parentheses end cell row blank equals cell 15 over 16 plus 26 over 4 end cell row blank equals cell fraction numerator 15 plus 104 over denominator 16 end fraction end cell row blank equals cell 119 over 16 end cell row blank equals cell 7 comma 4375 end cell row blank equals 7 end table 

Dengan demikian, a minus 2 b equals 7.

Oleh karena itu, jawaban yang benar adalah D.

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