Giải thích các bước giải:
\(\begin{array}{l}
1c,\\
\lim \sqrt n \left( {\sqrt {n + 1} - \sqrt n } \right)\\
= \lim \sqrt n .\frac{{n + 1 - n}}{{\sqrt {n + 1} + \sqrt n }}\\
= \lim \frac{{\sqrt n }}{{\sqrt {n + 1} + \sqrt n }}\\
= \lim \frac{1}{{\sqrt {1 + \frac{1}{n}} + 1}} = \frac{1}{{\sqrt 1 + 1}} = \frac{1}{2}\\
d,\\
\lim \frac{{1 - {{2.3}^n} + {6^n}}}{{{2^n}.\left( {{3^{n + 1}} - 5} \right)}}\\
= \lim \frac{{1 - {{2.3}^n} + {6^n}}}{{{{3.6}^n} - {{5.2}^n}}}\\
= \lim \frac{{{{\left( {\frac{1}{6}} \right)}^n} - 2.{{\left( {\frac{1}{2}} \right)}^n} + 1}}{{3 - 5.{{\left( {\frac{1}{3}} \right)}^n}}}\\
= \frac{1}{3}\\
2,\\
\lim \frac{{\sqrt {a{n^2} + 1} - \sqrt {4n - 2} }}{{5n + 2}} = 2\\
\Leftrightarrow \lim \frac{{\sqrt {a + \frac{1}{{{n^2}}}} - \sqrt {\frac{4}{n} - \frac{2}{{{n^2}}}} }}{{5 + \frac{2}{n}}} = 2\\
\Leftrightarrow \frac{{\sqrt a - 0}}{5} = 2\\
\Leftrightarrow \sqrt a = 10\\
\Leftrightarrow a = 100\\
3,\\
\lim \left( {\sqrt {{n^2} + an + 5} - \sqrt {{n^2} + bn + 3} } \right) = 2\\
\Leftrightarrow \lim \frac{{{n^2} + an + 5 - {n^2} - bn - 3}}{{\sqrt {{n^2} + an + 5} + \sqrt {{n^2} + bn + 3} }} = 2\\
\Leftrightarrow \lim \frac{{\left( {a - b} \right)n + 2}}{{\sqrt {{n^2} + an + 5} + \sqrt {{n^2} + bn + 3} }} = 2\\
\Leftrightarrow \lim \frac{{\left( {a - b} \right) + \frac{2}{n}}}{{\sqrt {1 + \frac{a}{n} + \frac{5}{{{n^2}}}} + \sqrt {1 + \frac{b}{n} + \frac{3}{{{n^2}}}} }} = 2\\
\Leftrightarrow \frac{{a - b}}{{\sqrt 1 + \sqrt 1 }} = 2\\
\Leftrightarrow a - b = 4
\end{array}\)
