Tính lim [n(căn bậc hai (n^2 + 2) - căn bậc hai (n^2 - 1)]
Câu hỏi:
Tính \(\lim \left[ {n\left( {\sqrt {{n^2} + 2} - \sqrt {{n^2} - 1} } \right)} \right]\).
Trả lời:
\(\lim \left[ {n\left( {\sqrt {{n^2} + 2} - \sqrt {{n^2} - 1} } \right)} \right]\)
\( = \lim \left[ {n.\frac{{\left( {\sqrt {{n^2} + 2} - \sqrt {{n^2} - 1} } \right)\left( {\sqrt {{n^2} + 2} + \sqrt {{n^2} - 1} } \right)}}{{\sqrt {{n^2} + 2} + \sqrt {{n^2} - 1} }}} \right]\)
\( = \lim \left[ {n.\frac{{\left( {{n^2} + 2} \right) - \left( {{n^2} - 1} \right)}}{{\sqrt {{n^2} + 2} + \sqrt {{n^2} - 1} }}} \right]\)
\( = \lim \frac{{3n}}{{\sqrt {{n^2} + 2} + \sqrt {{n^2} - 1} }}\)
\( = \lim \frac{3}{{\sqrt {1 + \frac{2}{{{n^2}}}} + \sqrt {1 - \frac{1}{{{n^2}}}} }}\)
\( = \lim \frac{3}{{\sqrt {1 + 0} + \sqrt {1 - 0} }}\)
\( = \frac{3}{2}\).
