Đáp án:
\[T = \frac{1}{8}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
T = \lim \left( {\sqrt {{{16}^{n + 1}} + {4^n}} - \sqrt {{{16}^{n + 1}} + {3^n}} } \right)\\
= \lim \frac{{{{16}^{n + 1}} + {4^n} - \left( {{{16}^{n + 1}} + {3^n}} \right)}}{{\sqrt {{{16}^{n + 1}} + {4^n}} + \sqrt {{{16}^{n + 1}} + {3^n}} }}\\
= \lim \frac{{{4^n} - {3^n}}}{{\sqrt {{{16}^{n + 1}} + {4^n}} + \sqrt {{{16}^{n + 1}} + {3^n}} }}\\
= \lim \frac{{1 - {{\left( {\frac{3}{4}} \right)}^n}}}{{\sqrt {\frac{{{{16}^{n + 1}} + {4^n}}}{{{{16}^n}}}} + \sqrt {\frac{{{{16}^{n + 1}} + {3^n}}}{{{{16}^n}}}} }}\\
= \lim \frac{{1 - {{\left( {\frac{3}{4}} \right)}^n}}}{{\sqrt {16 + {{\left( {\frac{1}{4}} \right)}^n}} + \sqrt {16 + {{\left( {\frac{3}{{16}}} \right)}^n}} }}\\
= \frac{1}{{\sqrt {16} + \sqrt {16} }} = \frac{1}{8}
\end{array}\)
