Question

$li{m}_{x\to 0}[1^{1/si{n}^{2}x}+2^{1/si{n}^{2}x}+....+n^{1/si{n}^{2}x}{]}^{si{n}^{2}x}$

• A. $\infty$
• B. 0
• C. $\frac{n+1}{2}$
• D. $n$

Answer 1

Answer: (D)

$n$

Let $\frac{1}{si{n}^{2}x}=y$
$li{m}_{x\to 0}[1^{1/si{n}^{2}x}+2^{1/si{n}^{2}x}+....+n^{1/si{n}^{2}x}{]}^{si{n}^{2}x}=li{m}_{y\to \infty }[1^y+2^y+....+n^y{]}^{1/y}=li{m}_{y\to \infty }n{({\left(\frac{1}{n}\right)}^y+{\left(\frac{2}{n}\right)}^y+....+{\left(\frac{n}{n}\right)}^y)}^{1/y}=n{(0+0+...+1)}^{1/y}=n$