Pertanyaan

Nilai dari x → 2 lim ( 2 − x 1 + x − x 2 + x + 3 ) adalah...

Nilai dari $x \to 2 lim (\frac{1 + x - x ^{2} + x + 3}{2 - x})$ adalah...

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E. Nur

Master Teacher

Mahasiswa/Alumni Institut Teknologi Sepuluh Nopember

Jawaban terverifikasi

Jawaban

nilai adalah .

nilai $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 1 plus x minus square root of x squared plus x plus 3 end root over denominator 2 minus x end fraction close parentheses end cell end table end style$ adalah

Pembahasan

Dengan konsep dasar limit diperoleh Akibatnya limit diatas mempunyai bentuk tak tentu . Sehingga dengan mengalikan akar sekawannya diperoleh Dengan demikian nilai adalah .

Dengan konsep dasar limit diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 2 of open parentheses fraction numerator 1 plus x plus square root of x squared plus x plus 3 end root over denominator 2 minus x end fraction close parentheses end cell equals cell limit as x rightwards arrow 2 of open parentheses fraction numerator 1 plus 2 plus square root of 2 squared plus 2 plus 3 end root over denominator 2 minus 2 end fraction close parentheses end cell row blank equals cell fraction numerator 3 minus square root of 9 over denominator 2 minus 2 end fraction end cell row blank equals cell fraction numerator 3 minus 3 over denominator 2 minus 2 end fraction end cell row blank equals cell 0 over 0 end cell end table end style$

Akibatnya limit diatas mempunyai bentuk tak tentu . Sehingga dengan mengalikan akar sekawannya diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of open parentheses fraction numerator 1 plus x minus square root of x squared plus x plus 3 end root over denominator 2 minus x end fraction close parentheses times open parentheses fraction numerator 1 plus x plus square root of x squared plus x plus 3 end root over denominator 1 plus x plus square root of x squared plus x plus 3 end root end fraction close parentheses end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator open parentheses 1 plus x close parentheses squared minus open parentheses x squared plus x plus 3 close parentheses over denominator 2 minus x end fraction times fraction numerator 1 over denominator 1 plus x plus square root of x squared plus x plus 3 end root end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator open parentheses x squared plus 2 x plus 1 close parentheses minus open parentheses x squared minus x minus 3 close parentheses over denominator open parentheses 2 minus x close parentheses open parentheses 1 plus x plus square root of x squared plus x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator x squared plus 2 x plus 1 minus x squared minus x minus 3 over denominator open parentheses 2 minus x close parentheses open parentheses 1 plus x plus square root of x squared plus x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator x minus 2 over denominator open parentheses 2 minus x close parentheses open parentheses 1 plus x plus square root of x squared plus x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator negative open parentheses 2 minus x close parentheses over denominator open parentheses 2 minus x close parentheses open parentheses 1 plus x plus square root of x squared plus x plus 3 end root close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 2 of fraction numerator negative 1 over denominator open parentheses 1 plus x plus square root of x squared plus x plus 3 end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 1 over denominator open parentheses 1 plus 2 plus square root of 2 squared plus 2 plus 3 end root close parentheses end fraction end cell row blank equals cell fraction numerator negative 1 over denominator open parentheses 3 plus square root of 9 close parentheses end fraction end cell row blank equals cell fraction numerator negative 1 over denominator 3 plus 3 end fraction end cell row blank equals cell negative 1 over 6 end cell end table end style$

Dengan demikian nilai $begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell limit as x rightwards arrow 2 of end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 1 plus x minus square root of x squared plus x plus 3 end root over denominator 2 minus x end fraction close parentheses end cell end table end style$ adalah .