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Câu hỏi: Tính giới hạn: $\lim \left[ \left( 1-\dfrac{1}{{{2}^{2}}} \right)\left( 1-\dfrac{1}{{{3}^{2}}} \right)...\left( 1-\dfrac{1}{{{n}^{2}}} \right) \right]$.
A. $\dfrac{1}{4}$.
B. $\dfrac{1}{2}$.
C. $1$.
D. $\dfrac{3}{2}$.
A. $\dfrac{1}{4}$.
B. $\dfrac{1}{2}$.
C. $1$.
D. $\dfrac{3}{2}$.
$\lim \left[ \left( 1-\dfrac{1}{{{2}^{2}}} \right)\left( 1-\dfrac{1}{{{3}^{2}}} \right)...\left( 1-\dfrac{1}{{{n}^{2}}} \right) \right]=\lim \left[ \left( 1-\dfrac{1}{2} \right)\left( 1+\dfrac{1}{2} \right)\left( 1-\dfrac{1}{3} \right)\left( 1+\dfrac{1}{3} \right)...\left( 1-\dfrac{1}{n} \right)\left( 1+\dfrac{1}{n} \right) \right]$
$=\lim \left( \dfrac{1}{2}.\dfrac{3}{2}.\dfrac{2}{3}.\dfrac{4}{3}...\dfrac{n-1}{n}.\dfrac{n+1}{n} \right)$ $=\lim \left( \dfrac{1}{2}.\dfrac{n+1}{n} \right)=\dfrac{1}{2}$.
$=\lim \left( \dfrac{1}{2}.\dfrac{3}{2}.\dfrac{2}{3}.\dfrac{4}{3}...\dfrac{n-1}{n}.\dfrac{n+1}{n} \right)$ $=\lim \left( \dfrac{1}{2}.\dfrac{n+1}{n} \right)=\dfrac{1}{2}$.
Đáp án B.