Tính: Lim 1/ 2 căn 1 + 1 căn 2

    Ta có \(\lim \left[ {\frac{1}{{2\sqrt 1 + 1\sqrt 2 }} + \frac{1}{{3\sqrt 2 + 2\sqrt 3 }} + \ldots + \frac{1}{{\left( {n + 1} \right)\sqrt n + n\sqrt {\left( {n + 1} \right)} }}} \right]\)               \( = \lim \left[ {\frac{{2\sqrt 1 - 1\sqrt 2 }}{{2.1}} + \frac{{3\sqrt 2 - 2\sqrt 3 }}{{3.2}} + \ldots + \frac{{\left( {n + 1} \right)\sqrt n - n\sqrt {n + 1} }}{{\left( {n + 1} \right).n}}} \right]\)\( = \lim \left[ {\sqrt 1 - \frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }} - \frac{1}{{\sqrt 3 }} + \ldots + \frac{1}{{\sqrt n }} - \frac{1}{{\sqrt {n + 1} }}} \right] = \lim \left[ {1 - \frac{1}{{\sqrt {n + 1} }}} \right] = 1\)