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Tentukanlah x→∞lim​3x−25x2−2x+1​4x2−1​+x2−1​+9x2+3x−1​​!

Pertanyaan

Tentukanlah limit as x rightwards arrow infinity of fraction numerator square root of 4 x squared minus 1 end root plus square root of x squared minus 1 end root plus square root of 9 x squared plus 3 x minus 1 end root over denominator 3 x minus square root of 25 x squared minus 2 x plus 1 end root end fraction

Y. Herlanda

Master Teacher

Mahasiswa/Alumni STKIP PGRI Jombang

Jawaban terverifikasi

Jawaban

nilai dari limit as x rightwards arrow infinity of fraction numerator square root of 4 x squared minus 1 end root plus square root of x squared minus 1 end root plus square root of 9 x squared plus 3 x minus 1 end root over denominator 3 x minus square root of 25 x squared minus 2 x plus 1 end root end fraction equals negative 3.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold minus bold 3.

Limit Bentuk Tak Tentu bold infinity over bold infinity 

limit as x rightwards arrow infinity of fraction numerator square root of 4 x squared minus 1 end root plus square root of x squared minus 1 end root plus square root of 9 x squared plus 3 x minus 1 end root over denominator 3 x minus square root of 25 x squared minus 2 x plus 1 end root end fraction

Substitusikan nilai x equals infinity ke fungsi di atas!

fraction numerator square root of 4 open parentheses infinity close parentheses squared minus 1 end root plus square root of open parentheses infinity close parentheses squared minus 1 end root plus square root of 9 open parentheses infinity close parentheses squared plus 3 open parentheses infinity close parentheses minus 1 end root over denominator 3 open parentheses infinity close parentheses minus square root of 25 open parentheses infinity close parentheses squared minus 2 open parentheses infinity close parentheses plus 1 end root end fraction equals infinity over infinity 

Karena infinity over infinity merupakan bentuk tak tentu, maka limit fungsi di atas perlu kita ubah dengan cara mengeluarkan faktor x dari betuk tersebut. Sehingga diperoleh:

begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow infinity of fraction numerator square root of 4 x squared minus 1 end root plus square root of x squared minus 1 end root plus square root of 9 x squared plus 3 x minus 1 end root over denominator 3 x minus square root of 25 x squared minus 2 x plus 1 end root end fraction end cell equals cell limit as x rightwards arrow infinity of fraction numerator square root of x squared open parentheses 4 minus begin display style 1 over x squared end style close parentheses end root plus square root of x squared open parentheses 1 minus begin display style 1 over x squared end style close parentheses end root plus square root of x squared open parentheses 9 plus begin display style 3 over x end style minus begin display style 1 over x squared end style close parentheses end root over denominator x open parentheses 3 close parentheses minus square root of x squared open parentheses 25 minus begin display style 2 over x end style plus begin display style 1 over x end style close parentheses end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator x square root of open parentheses 4 minus begin display style 1 over x squared end style close parentheses end root plus x square root of open parentheses 1 minus begin display style 1 over x squared end style close parentheses end root plus x square root of open parentheses 9 plus begin display style 3 over x end style minus begin display style 1 over x squared end style close parentheses end root over denominator x open parentheses 3 close parentheses minus x square root of open parentheses 25 minus begin display style 2 over x end style plus begin display style 1 over x end style close parentheses end root end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator up diagonal strike x open parentheses square root of open parentheses 4 minus 1 over x squared close parentheses end root plus square root of open parentheses 1 minus 1 over x squared close parentheses end root plus square root of open parentheses 9 plus 3 over x minus 1 over x squared close parentheses end root close parentheses over denominator up diagonal strike x open parentheses 3 minus square root of open parentheses 25 minus 2 over x plus 1 over x close parentheses end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow infinity of fraction numerator square root of open parentheses 4 minus 1 over x squared close parentheses end root plus square root of open parentheses 1 minus 1 over x squared close parentheses end root plus square root of open parentheses 9 plus 3 over x minus 1 over x squared close parentheses end root over denominator 3 minus square root of open parentheses 25 minus 2 over x plus 1 over x close parentheses end root end fraction end cell row blank equals cell fraction numerator square root of open parentheses 4 minus 1 over infinity squared close parentheses end root plus square root of open parentheses 1 minus 1 over infinity squared close parentheses end root plus square root of open parentheses 9 plus 3 over infinity minus 1 over infinity squared close parentheses end root over denominator 3 minus square root of open parentheses 25 minus 2 over infinity plus 1 over infinity close parentheses end root end fraction end cell row blank equals cell fraction numerator square root of 4 minus 0 end root plus square root of 1 minus 0 end root plus square root of 9 plus 0 minus 0 end root over denominator 3 minus square root of 25 minus 0 plus 0 end root end fraction end cell row blank equals cell fraction numerator square root of 4 plus square root of 1 plus square root of 9 over denominator 3 minus square root of 25 end fraction end cell row blank equals cell fraction numerator 2 plus 1 plus 3 over denominator 3 minus 5 end fraction end cell row blank equals cell fraction numerator 6 over denominator negative 2 end fraction end cell row blank equals cell negative 3 end cell end table end style 

Dengan demikian, nilai dari limit as x rightwards arrow infinity of fraction numerator square root of 4 x squared minus 1 end root plus square root of x squared minus 1 end root plus square root of 9 x squared plus 3 x minus 1 end root over denominator 3 x minus square root of 25 x squared minus 2 x plus 1 end root end fraction equals negative 3.

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