Nilai dari x→∞lim 9x2+6x−2−3x+1 adalah ...

Pertanyaan

Nilai dari $\lim_{x\rightarrow\infty}\;\sqrt{9x^2+6x-2}-3x+1$ adalah ...

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

hasil dari limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus 3 x plus 1

adalah

Pembahasan

Hasil dari limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus 3 x plus 1 :

Jika terdapat limit takhingga bentuk akar dimana fungsi di dalam akar merupakan suatu persamaan kuadrat, maka berlaku:

$begin mathsize 12px style limit as x rightwards arrow infinity of space open parentheses square root of a x squared plus b x plus c end root minus square root of p x squared plus q x plus r end root close parentheses equals open curly brackets table attributes columnalign left end attributes row cell infinity comma space untuk space a greater than p end cell row cell fraction numerator b minus q over denominator 2 square root of a end fraction comma space untuk space a equals p end cell row cell negative infinity comma space untuk space a less than p end cell end table close end style$

karena a equals p equals 9 , maka:

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

Atau dapat menggunakan cara lainnya yaitu:

$limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus 3 x plus 1 equals limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus open parentheses 3 x minus 1 close parentheses equals limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus square root of open parentheses 3 x minus 1 close parentheses squared end root equals limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus square root of 9 x squared minus 6 x plus 1 end root equals limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus square root of 9 x squared minus 6 x plus 1 end root space cross times fraction numerator square root of 9 x squared plus 6 x minus 2 end root plus square root of 9 x squared minus 6 x plus 1 end root over denominator square root of 9 x squared plus 6 x minus 2 end root plus square root of 9 x squared minus 6 x plus 1 end root end fraction equals limit as x rightwards arrow infinity of space fraction numerator open parentheses 9 x squared plus 6 x minus 2 close parentheses minus open parentheses 9 x squared minus 6 x plus 1 close parentheses over denominator square root of 9 x squared plus 6 x minus 2 end root plus square root of 9 x squared minus 6 x plus 1 end root end fraction equals limit as x rightwards arrow infinity of space fraction numerator 12 x minus 3 over denominator square root of 9 x squared plus 6 x minus 2 end root plus square root of 9 x squared minus 6 x plus 1 end root end fraction equals limit as x rightwards arrow infinity of space fraction numerator begin display style fraction numerator 12 x over denominator x end fraction end style minus begin display style 3 over x end style over denominator square root of begin display style fraction numerator 9 x squared over denominator x squared end fraction end style plus begin display style fraction numerator 6 x over denominator x squared end fraction end style minus begin display style 2 over x squared end style end root plus square root of begin display style fraction numerator 9 x squared over denominator x squared end fraction end style minus begin display style fraction numerator 6 x over denominator x squared end fraction end style plus begin display style 1 over x squared end style end root end fraction equals limit as x rightwards arrow infinity of space fraction numerator begin display style 12 minus 3 over x end style over denominator square root of begin display style 9 plus 6 over x minus 2 over x squared end style end root plus square root of begin display style 9 minus 6 over x plus 1 over x squared end style end root end fraction equals fraction numerator begin display style 12 minus 0 end style over denominator square root of begin display style 9 plus 0 minus 0 end style end root plus square root of begin display style 9 minus 0 plus 0 end style end root end fraction equals fraction numerator 12 over denominator square root of 9 plus square root of 9 end fraction equals fraction numerator 12 over denominator 3 plus 3 end fraction equals 12 over 6 equals 2$

Jadi, hasil dari limit as x rightwards arrow infinity of space square root of 9 x squared plus 6 x minus 2 end root minus 3 x plus 1 adalah .