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

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

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Perhatikan bahwa Sehingga, Karena , maka . Ketika maka

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begin mathsize 14px style stack bold l bold i bold m with bold x bold rightwards arrow bold infinity below begin bold style left parenthesis 1 minus 1 over n right parenthesis end style to the power of bold n bold equals bold e to the power of bold minus bold 1 end exponent end style

Sehingga,

$begin mathsize 14px style limit as straight x rightwards arrow infinity of open parentheses fraction numerator straight x minus 3 over denominator straight x minus 1 end fraction close parentheses to the power of straight x minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses fraction numerator straight x minus 1 minus 2 over denominator straight x minus 1 end fraction close parentheses to the power of straight x minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses fraction numerator straight x minus 1 over denominator straight x minus 1 end fraction minus fraction numerator 2 over denominator straight x minus 1 end fraction close parentheses to the power of straight x minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses 1 minus fraction numerator 2 over denominator straight x minus 1 end fraction close parentheses to the power of straight x minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses 1 minus fraction numerator 1 over denominator begin display style fraction numerator straight x minus 1 over denominator 2 end fraction end style end fraction close parentheses to the power of straight x minus 1 end exponent end style$

Karena $begin mathsize 14px style straight n equals fraction numerator straight x minus 1 over denominator 2 end fraction end style$ , maka . Ketika maka

$begin mathsize 14px style limit as straight x rightwards arrow infinity of open parentheses fraction numerator straight x minus 3 over denominator straight x minus 1 end fraction close parentheses to the power of straight x minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses 1 minus fraction numerator 1 over denominator begin display style fraction numerator straight x minus 1 over denominator 2 end fraction end style end fraction close parentheses to the power of straight x minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses 1 minus 1 over straight n close parentheses to the power of open parentheses 2 straight n plus 1 close parentheses minus 1 end exponent equals limit as straight x rightwards arrow infinity of open parentheses 1 minus 1 over straight n close parentheses to the power of 2 straight n end exponent equals limit as straight x rightwards arrow infinity of open parentheses open parentheses 1 minus 1 over straight n close parentheses to the power of straight n close parentheses squared equals open parentheses straight e to the power of negative 1 end exponent close parentheses squared equals 1 over straight e squared end style$