lim-n-a-1-n-b-1-n-2-n-a-b-R-

Question Number 127961 by bramlexs22 last updated on 03/Jan/21

\lim_{n \to \infty} {(\frac{\sqrt[n]{a} + \sqrt[n]{b}}{2})}^{n} = ?

a, b \in R

Answered by liberty last updated on 03/Jan/21

\lim_{n \to \infty} {(\frac{\sqrt[n]{a} + \sqrt[n]{b}}{2})}^{n} = e^{\lim_{n \to \infty} \ln {(\frac{\sqrt[n]{a} + \sqrt[n]{b}}{2})}^{n}}

= e^{\lim_{x \to 0^{+}} (\frac{\ln (a^{x} + b^{x}) - \ln 2}{x})}; where n = \frac{1}{x}

= e^{\lim_{x \to 0^{+}} (\frac{(\ln a) . a^{x} + (\ln b) . b^{x}}{a^{x} + b^{x}})}

= e^{(\frac{\ln a + \ln b}{2})} = e^{\ln {(ab)}^{\frac{1}{2}}} = {(ab)}^{\frac{1}{2}} = \sqrt{ab}

Facebook Tweet Pin