Cho dãy số như sau (({u_n})) với ({u_n} = (n – 1)sqrt {dfrac{{2n + 2}}{{{n^4} + {n^2} – 1}}} ).

(begin{array}{l}lim {u_n} = lim left( {(n – 1)sqrt {dfrac{{2n + 2}}{{{n^4} + {n^2} – 1}}} } right) \= lim sqrt {dfrac{{left( {2n + 2} right){{left( {n – 1} right)}^2}}}{{{n^4} + {n^2} – 1}}} \ = lim sqrt {dfrac{{left( {2n + 2} right)left( {{n^2} – 2n + 1} right)}}{{{n^4} + {n^2} – 1}}} \ = lim sqrt {dfrac{{2{n^3} – 6{n^2} – 2}}{{{n^4} + {n^2} – 1}}}  \= lim sqrt {dfrac{{dfrac{2}{n} – dfrac{6}{{{n^2}}} – dfrac{2}{{{n^4}}}}}{{1 + dfrac{1}{{{n^2}}} – dfrac{1}{{{n^4}}}}}}  \= sqrt {dfrac{0}{1}}  = 0end{array})