\(A-B=\dfrac{10^8+1}{10^9+1}-\dfrac{10^9+1}{10^{10}+1}=\dfrac{\left(10^8+1\right)\left(10^{10}+1\right)-\left(10^9+1\right)^2}{\left(10^9+1\right)\left(10^{10}+1\right)}\)
\(\Rightarrow A-B=\dfrac{10^{18}+10^{10}+10^8+1-\left(10^8+2.10^9+1\right)}{\left(10^9+1\right)\left(10^{10}+1\right)}\)
\(=\dfrac{10^{18}+10^{10}+10^8+1-10^8-2.10^9-1}{\left(10^9+1\right)\left(10^{10}+1\right)}\)
\(=\dfrac{10^{10}+10^8-2.10^9}{\left(10^9+1\right)\left(10^{10}+1\right)}=\dfrac{10^8\left(10^2+1-2.10\right)}{\left(10^9+1\right)\left(10^{10}+1\right)}=\dfrac{10^8.81}{\left(10^9+1\right)\left(10^{10}+1\right)}>0\)
\(\Rightarrow A-B>0\)
\(\Rightarrow A>B\)
