smc

We have $\sqrt{65}>8>7.5$. Also $7.5^2=(7+0.5{)}^2=7^2+2\cdot 7\cdot 0.5+0.5^2=49+7+0.25=56.25<63$, so $\sqrt{63}>7.5$. Thus $\sqrt{65}+\sqrt{63}>7.5+7.5=15$. Now notice that $\sqrt{65}-\sqrt{63}=\frac{(\sqrt{65}-\sqrt{63})(\sqrt{65}+\sqrt{63})}{\sqrt{65}+\sqrt{63}}=\frac{2}{\sqrt{65}+\sqrt{63}}$, so $\sqrt{65}-\sqrt{63}<\frac{2}{15}=\mathrm{0.1333333...}$, so the answer must be $A$ or $B$. To determine which, we write $\sqrt{65}-\sqrt{63}>0.125⟺65-2\sqrt{65\cdot 63}+63>0.015625⟺128-0.015625>2\sqrt{4095}⟺\sqrt{4095}<64-0.0078125⟺4095<4096-128\cdot 0.0078125+0.0078125^2=4096-1+0.0078125^2$ which is true. Hence as the expression is greater than $0.125$, and less than or equal to $0.13$ (since we showed it is certainly less than $\mathrm{0.1333333...}$), it is closest to $0.13$, which is answer $\boxed{.13}$. ($\sqrt{65}-\sqrt{63}$ is approximately equal to $0.125003815$)