Pertanyaan

x → 3 lim x − 4 x − 3 5 x − 15 = …

$x \to 3 lim \frac{5 x - 15}{x - 4 x - 3} = \dots$

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Jawaban terverifikasi

Jawaban

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$begin mathsize 14px style limit as x rightwards arrow 3 of fraction numerator 5 x minus 15 over denominator x minus square root of 4 x minus 3 end root end fraction equals 15 end style$ .

Pembahasan

Jika kita melakukan substitusi langsung pada permasalahan limit di atas, kita akan memperoleh bentuk tak tentu . Oleh karena itu, kita akan menyelesaikan masalah limit tersebut dengan cara mengalikan akar sekawan penyebutnya. Perhatikan bahwa Dengan demikian, .

Jika kita melakukan substitusi langsung pada permasalahan limit di atas, kita akan memperoleh bentuk tak tentu . Oleh karena itu, kita akan menyelesaikan masalah limit tersebut dengan cara mengalikan akar sekawan penyebutnya.

Perhatikan bahwa

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell limit as x rightwards arrow 3 of fraction numerator 5 x minus 15 over denominator x minus square root of 4 x minus 3 end root end fraction end cell equals cell limit as x rightwards arrow 3 of fraction numerator 5 x minus 15 over denominator x minus square root of 4 x minus 3 end root end fraction times fraction numerator x plus square root of 4 x minus 3 end root over denominator x plus square root of 4 x minus 3 end root end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator open parentheses 5 x minus 15 close parentheses open parentheses x plus square root of 4 x minus 3 end root close parentheses over denominator x squared minus open parentheses square root of 4 x minus 3 end root close parentheses squared end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator open parentheses 5 x minus 15 close parentheses open parentheses x plus square root of 4 x minus 3 end root close parentheses over denominator x squared minus open parentheses 4 x minus 3 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator open parentheses 5 x minus 15 close parentheses open parentheses x plus square root of 4 x minus 3 end root close parentheses over denominator x squared minus 4 x plus 3 end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator 5 up diagonal strike open parentheses x minus 3 close parentheses end strike open parentheses x plus square root of 4 x minus 3 end root close parentheses over denominator up diagonal strike open parentheses x minus 3 close parentheses end strike open parentheses x minus 1 close parentheses end fraction end cell row blank equals cell limit as x rightwards arrow 3 of fraction numerator 5 open parentheses x plus square root of 4 x minus 3 end root close parentheses over denominator open parentheses x minus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 5 open parentheses 3 plus square root of 4 times 3 minus 3 end root close parentheses over denominator open parentheses 3 minus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 5 open parentheses 3 plus square root of 9 close parentheses over denominator open parentheses 2 close parentheses end fraction end cell row blank equals cell fraction numerator 5 open parentheses 3 plus 3 close parentheses over denominator 2 end fraction end cell row blank equals 15 end table end style$

Dengan demikian, $begin mathsize 14px style limit as x rightwards arrow 3 of fraction numerator 5 x minus 15 over denominator x minus square root of 4 x minus 3 end root end fraction equals 15 end style$ .