limx→1(xn-1n(x-1))1x-1=

Ltx→1(xn-1n(x-1))1x-1 1∞ form =eLtx→1 xn-1-nx-nn(x-1)2 =eLtx→1n(xn-1)-n2n(x-1) =eLtx→1n(n-1)xn-22n =en-12

limx→1(xn-1n(x-1))1x-1=

Ltx→1(xn-1n(x-1))1x-1 1∞ form =eLtx→1 xn-1-nx-nn(x-1)2 =eLtx→1n(xn-1)-n2n(x-1) =eLtx→1n(n-1)xn-22n =en-12

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${\lim_{x \to 1} (\frac{x^{n} - 1}{n (x - 1)})}^{\frac{1}{x - 1}}$ =

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$e^{\frac{n - 1}{2}}$

$e^{\frac{1 + n}{2}}$

$e^{\frac{n}{2}}$

$e^{\frac{1 - n}{2}}$

answer is D.

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Detailed Solution

${\underset{x \to 1}{Lt} (\frac{x^{n} - 1}{n (x - 1)})}^{\frac{1}{x - 1}} (1^{\infty} form) = e^{\underset{x \to 1}{Lt}} \frac{x^{n} - 1 - nx - n}{n {(x - 1)}^{2}} = e^{\underset{x \to 1}{Lt}} \frac{n (x^{n - 1}) - n}{2 n (x - 1)} = e^{\underset{x \to 1}{Lt}} \frac{n (n - 1) x^{n - 2}}{2 n} = e^{\frac{n - 1}{2}}$