[latex] \lim_{n \to \infty} \frac{( \sqrt{n} + \sqrt[3]{n} + \sqrt[4]{n} )}{ \sqrt{2n +1} }[/latex]

[latex] \lim_{n \to \infty} \frac{ ( \sqrt{n} + \sqrt[3]{n} + \sqrt[4]{n} )}{ \sqrt{2n +1} }= \lim_{n \to \infty} \frac{ \sqrt{n} ( 1 + n^{ \frac{1}{3}- \frac{1}{2} } + n^{ \frac{1}{4}- \frac{1}{2} } )}{ \sqrt{n} \sqrt{2 +\frac{1}{n} } }=\\\\=\lim_{n \to \infty} \frac{ ( 1 + n^{ \frac{1}{3}- \frac{1}{2} } + n^{ \frac{1}{4}- \frac{1}{2} } )}{ \sqrt{2 +\frac{1}{n} } }= \frac{1}{ \sqrt{2} } [/latex]