a) \(\lim \dfrac{6n - 1}{3n +2}\) \(= \lim\dfrac{6 - \dfrac{1}{n}}{3 +\dfrac{2}{n}}\) = \(\dfrac{6}{3} = 2\).
b) \(\lim \dfrac{3n^{2}+n-5}{2n^{2}+1}\)\( = \lim \dfrac{3 +\dfrac{1}{n}-\dfrac{5}{n^{2}}}{2+\dfrac{1}{n^{2}}}= \dfrac{3}{2}\).
c) Chia cả tử và mẫu của phân thức cho \(4^n\) ta được:
\(\lim \dfrac{3^{n}+5.4^{n}}{4^{n}+2^{n}}\) \(= \lim \dfrac{{\left( {{3 \over 4}} \right)^n}+5}{1+{\left( {{1 \over 2}} \right)^n}}\) \(=\dfrac{0+5}{1+0}=\dfrac{5}{1}\) \(= 5\).
d) \(\lim \dfrac{\sqrt{9n^{2}-n+1}}{4n -2}\) = \(\lim \dfrac{\sqrt{{n^2}\left( {9 - {1 \over n} + {1 \over {{n^2}}}} \right)}}{n(4-\dfrac{2}{n})}\)= \(\lim \dfrac{\sqrt{9-\dfrac{1}{n}+\dfrac{1}{n^{2}}}}{4-\dfrac{2}{n}}\) =\(\dfrac{\sqrt{9}}{4}\)= \(\dfrac{3}{4}\).
