Question

PLsss i need Help!

lim ( (2^n + 3^n)/ (5n-1) ) ^ 1/n
n goes infinty

Answer 1

\[(\frac{2^n+3^n}{5n-1})^\frac{1}{n}\]

Answer 2

\[A=\lim_{x \rightarrow \infty} \left\{ (2^{x}+3^{x})/(5x-1)\right\}^{1/x}\]

\[\ln A=\lim_{x \rightarrow \infty}(1/x)*\ln \left\{ (2^{x}+3^{x})/(5x-1)\right\}\]
\[=\lim_{x \rightarrow \infty}\ln \left\{ (2^{x}+3^{x})/(5x-1)\right\}/x\]
this is \[\infty/\infty\]


\[=\lim_{x \rightarrow \infty}\ln \left\{ (2^{x}+3^{x})/x-\ln(5x-1)\right\}/x\]
use l'Hospital's
\[(2^{x}\ln2+3^{x}\ln3)/(2^{x}+3^{x})-5/(5x-1)\]

\[\lim_{x \rightarrow \infty}(2^{x}\ln2)/(2^{x}+3^{x})+\lim_{x \rightarrow \infty}(3^{x}\ln3)/(2^{x}+3^{x})-\lim_{x \rightarrow \infty}5/(5x-1)\]

first term is zero, second term is ln3 and third term is zero

so lnA=ln3

A=3

Answer 3

thx a lot good solving