Question

Find $m+n$

$\underset{x\to \infty }{lim}\frac{2\sqrt{x}+3\sqrt[3]{3x}+5\sqrt[5]{5x}}{\sqrt{(5x-2)}+\sqrt[3]{3(3x-2)}}$ is $\frac{m}{\sqrt{n}}$

Answer 1

${lim}_{x\to \infty }\frac{2x^{\frac{1}{2}}+3x^{\frac{1}{3}}+5x^{\frac{1}{5}}}{\sqrt{x(5-\frac{2}{x})}+\sqrt[3]{3x(3-\frac{2}{x})}}$

$={lim}_{x\to \infty }\frac{x^{\frac{1}{2}}(2+3x^{\frac{1}{3}-\frac{1}{2}}+5x^{\frac{1}{5}-\frac{1}{2}})}{x^{\frac{1}{2}}{(5-\frac{2}{x})}^{\frac{1}{2}}+x^{\frac{1}{3}}{(3-\frac{2}{x})}^{\frac{1}{3}}}$

$={lim}_{x\to \infty }\frac{x^{\frac{1}{2}}(2+3x^{\frac{1}{3}-\frac{1}{2}}+5x^{\frac{1}{5}-\frac{1}{2}})}{x^{\frac{1}{2}}[{(5-\frac{2}{x})}^{\frac{1}{2}}+x^{\frac{1}{3}-\frac{1}{2}}{(3-\frac{2}{x})}^{\frac{1}{3}}]}$

$={lim}_{x\to \infty }\frac{2+3x^{\frac{2-3}{6}}+5x^{\frac{2-5}{10}}}{{(5-\frac{2}{x})}^{\frac{1}{2}}+x^{\frac{2-3}{6}}{(3-\frac{2}{x})}^{\frac{1}{3}}}$

$={lim}_{x\to \infty }\frac{2+3x^{\frac{-1}{6}}+5x^{\frac{-3}{10}}}{{(5-\frac{2}{x})}^{\frac{1}{2}}+x^{\frac{-1}{6}}{(3-\frac{2}{x})}^{\frac{1}{3}}}$

$={lim}_{x\to \infty }\frac{2+\frac{3}{x^{\frac{1}{6}}}+\frac{5}{x^{\frac{3}{10}}}}{{(5-\frac{2}{x})}^{\frac{1}{2}}+\frac{1}{x^{\frac{1}{6}}}{(3-\frac{2}{x})}^{\frac{1}{3}}}$

As $x\to \infty ,\frac{1}{x}\to 0$

$={lim}_{x\to \infty }\frac{2+0+0}{{(5-0)}^{\frac{1}{2}}+0{(3-0)}^{\frac{1}{3}}}$

$={lim}_{x\to \infty }\frac{2}{\sqrt{5}}=\frac{m}{\sqrt{n}}$(given)

where $m=2,n=5$

$\therefore m+n=2+5=7$