$\lim \left[\dfrac{1}{1.3}+\dfrac{1}{2.4}+\dfrac{1}{3.5}+\cdots+\dfrac{1}{n(n+2)}\right]\\ =\lim \dfrac{1}{2}\left[\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{3}-\dfrac{1}{5}+\cdots+\dfrac{1}{n}-\dfrac{1}{n-2}\right]\\ =\lim \dfrac{1}{2}\left[\dfrac{1}{1}+\dfrac{1}{2}-\dfrac{1}{n}-\dfrac{1}{n-2}\right]\\ =\lim \dfrac{1}{2}\left[\dfrac{3(n-2)n-2(n-2)-2n}{2(n-2)n}\right]\\ =\lim \left[\dfrac{3n^2-6n-2n+4-2n}{4(n-2)(n)}\right]\\ =\lim \left(\dfrac{3n^2-10n+4}{4n^2-8n}\right)\\ =\lim \left(\dfrac{3-\dfrac{10}{n}+\dfrac{4}{n^2}}{4-\dfrac{8}{n}}\right)\\ =\dfrac{3}{4}$
