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Jika limit as x rightwards arrow infinity of 3 x minus 5 minus square root of 9 x squared plus 2 a x plus b end root equals 2, maka nilai a adalah ...

N. Puspita

Master Teacher

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nilai a adalah negative 21.

Pembahasan

Terdapat beberapa cara untuk menentukan limit menuju tak hingga dengan bentuk akar, seperti berikut ini:

limit as x rightwards arrow infinity of open parentheses square root of a x squared plus b x plus c end root minus square root of p x squared plus q x plus r end root close parentheses equals open curly brackets table attributes columnalign left end attributes row infinity row cell fraction numerator b minus q over denominator 2 square root of a end fraction end cell row cell negative infinity end cell end table close  

Diperoleh begin mathsize 14px style infinity end style apabila a greater than p, diperoleh fraction numerator b minus q over denominator 2 square root of a end fraction untuk a equals p, dan begin mathsize 14px style negative infinity end style untuk a less than b. Uraikan bentuk limit as x rightwards arrow infinity of 3 x minus 5 minus square root of 9 x squared plus 2 a x plus b end root equals 2 terlebih dahulu seperti berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of 3 x minus 5 minus square root of 9 x squared plus 2 a x plus b end root end cell equals 2 row cell limit as x rightwards arrow infinity of open parentheses 3 x minus 5 close parentheses minus square root of 9 x squared plus 2 a x plus b end root end cell equals 2 row cell limit as x rightwards arrow infinity of square root of open parentheses 3 x minus 5 close parentheses squared end root minus square root of 9 x squared plus 2 a x plus b end root end cell equals 2 row cell limit as x rightwards arrow infinity of square root of 9 x squared minus 30 x plus 25 end root minus square root of 9 x squared plus 2 a x plus b end root end cell equals 2 end table 

Oleh karena a equals p maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator b minus q over denominator 2 square root of a end fraction end cell equals 2 row cell fraction numerator negative 30 minus 2 a over denominator 2 square root of 9 end fraction end cell equals 2 row cell fraction numerator negative 30 minus 2 a over denominator 2 open parentheses 3 close parentheses end fraction end cell equals 2 row cell fraction numerator negative 30 minus 2 a over denominator 6 end fraction end cell equals 2 row cell negative 30 minus 2 a end cell equals cell 2 times 6 end cell row cell negative 30 minus 2 a end cell equals 12 row cell negative 2 a end cell equals cell 12 plus 30 end cell row cell negative 2 a end cell equals 42 row a equals cell negative 21 end cell end table 

Dengan demikian nilai a adalah negative 21.

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