$\lim\Big( \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{n(n+1)}\Big)$
$=\lim\Big( 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{n}-\dfrac{1}{n+1}\Big)$
$=\lim\Big( 1-\dfrac{1}{n+1}\Big)$
$=\lim\dfrac{n+1-1}{n+1}$
$=\lim\dfrac{n}{n+1}$
$=\lim\dfrac{1}{1+\dfrac{1}{n}}$
$=1$