Question

Let $x_n=(2^n+3^n{)}^{1/2n}$ for all natural numbers n. Then.

• A. $\underset{n\to \infty }{lim}{x}_n=\infty$
• B. $\underset{n\to \infty }{lim}{x}_n=\sqrt{3}$
• C. $\underset{n\to \infty }{lim}{x}_n=\sqrt{3}+\sqrt{2}$
• D. $\underset{n\to \infty }{lim}{x}_n=\sqrt{5}$

Answer 1

The correct option is B $\underset{n\to \infty }{lim}{x}_n=\sqrt{3}$

$\underset{n\to \infty }{lim}(2^n+3^n{)}^{1/2n}$

$=\underset{n\to \infty }{lim}{3}^{n/2n}(1+(\frac{2}{3}{)}^n{)}^{1/2n}$

$=3^{1/2}$

$=\sqrt{3}$