tính giới hạn: lim[91-1/2^2)(1-1/3^)…(1-1/n^2) A.1 B1/4 C3/2 D1/2

Đáp án: D

Giải thích các bước giải:

$\lim\left(1-\dfrac{1}{2^2}\right)\left(1-\dfrac{1}{3^2}\right)….\left(1-\dfrac{1}{n^2}\right)$

$=\lim\dfrac{2^2-1}{2^2}.\dfrac{3^2-1}{3^2}…\dfrac{n^2-1}{n^2}$

$=\lim\dfrac{\left(2-1\right)\left(2+1\right)}{2^2}.\dfrac{\left(3-1\right)\left(3+1\right)}{3^2}…\dfrac{\left(n-1\right)\left(n+1\right)}{n^2}$

$=\lim\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}…\dfrac{\left(n-1\right)\left(n+1\right)}{n^2}$

$=\lim\dfrac{1.3}{2^2}.\dfrac{2.4}{3^2}.\dfrac{3.5}{4^2}…\dfrac{\left(n-1\right)\left(n+1\right)}{n^2}$

$=\lim\dfrac{n+1}{2n}$

$=\lim\dfrac12+\dfrac1{2n}$

$=\dfrac12$