limx→0 1x+2x+3x+…+nxn1x is equal to - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

A
$(n!)^{n}$
B
$(n!)^{\frac{1}{n}}$
C
$n!$
D
$\ln (n!)$

Solution:

$= lim_{x \to 0} {(\frac{1^{x} + 2^{x} + \dots . + n^{x}}{n})}^{\frac{1}{x}} = e^{lim_{x \to 0} \frac{1}{x} (\frac{1^{x} + 2^{x} + \dots . + n^{x}}{n} - 1)}$

$= e^{lim_{x \to 0} \frac{1}{x} (\frac{1^{x} - 1}{n} + \frac{2^{x} - 1}{n} + \dots + \frac{n^{x} - 1}{n}) A p p l y L' h o s p i t a l r u l e} = e^{\frac{1}{n} [\log 1 + \log 2 + \dots + \log n]} = {(n!)}^{\frac{1}{n}}$

$lim_{x \to 0} {(\frac{1^{x} + 2^{x} + 3^{x} + \dots + n^{x}}{n})}^{\frac{1}{x}}$ is equal to

Solution:

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