Question

GIven $0<a<b$, the value of $\underset{n\to \infty }{lim}{(a^n+b^n)}^{1/n}$ is:

• A. $a$
• B. $b$
• C. $\frac{1}{a}$
• D. $\frac{1}{b}$

Answer 1

The correct option is B $b$

$\underset{n\to \infty }{lim}(a^n+b^n{)}^{1/n}=\underset{n\to \infty }{lim}\left(\exp \left(\log \left(\right(a^n+b^n{)}^{1/n})\right)\right)=\underset{n\to \infty }{lim}\left(\exp \left(\frac{\log (a^n+b^n)}{n}\right)\right)=\exp \underset{n\to \infty }{lim}\left(\frac{\log (a^n+b^n)}{n}\right)=\exp \underset{n\to \infty }{lim}\left(\frac{\log \left(a\right)a^n+\log \left(b\right)b^n}{a^n+b^n}\right)=\exp \left(\log \right(a\left)\underset{n\to \infty }{lim}\left(\frac{a^n}{a^n+b^n}\right)+\log \left(4\right)\underset{n\to \infty }{lim}\left(\frac{b^n}{a^n+b^n}\right)\right)=\exp \left(\log \right(a){lim}_{n\to \infty }b\left(\frac{{\left(\frac{a}{b}\right)}^n}{{\left(\frac{a}{b}\right)}^n+1}\right)+\log (b\left)\underset{n\to \infty }{lim}\left(\frac{1}{{\left(\frac{a}{b}\right)}^n+1}\right)\right)=\exp \left(\log \right(a\left)\right(0)+\log (b\left)\right(1\left)\right)=b$

Hence, option 'B' is correct.