Question

The value of the limit $\underset{x\to 0}{lim}\{1^{1/si{n}^{2}x}+2^{1/si{n}^{2}x}+\dots \dots +n^{1/si{n}^{2}x}{\}}^{si{n}^{2}x}$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

$\underset{x\to 0}{lim}{\{1^{\frac{1}{si{n}^{2}x}}+2^{\frac{1}{si{n}^{2}x}}+\dots \dots +n^{\frac{1}{si{n}^{2}x}}\}}^{si{n}^{2}x}Put\frac{1}{si{n}^{2}x}=t\ge 1\therefore \underset{t\to \infty }{lim}(1^t+2^t+\dots \dots +n^t{)}^{\frac{1}{t}}=\underset{t\to \infty }{lim}(n^t{)}^{\frac{1}{t}}{[{\left(\frac{1}{n}\right)}^t+{\left(\frac{2}{n}\right)}^t+\dots \dots +1]}^{\frac{1}{t}}=n\underset{t\to \infty }{lim}{[{\left(\frac{1}{n}\right)}^t+{\left(\frac{2}{n}\right)}^t+\dots \dots +1]}^{\frac{1}{t}}=n[0+0+\dots \dots +1{]}^0=n$