What are the order of the reaction under the following conditions?
Let's compare the rate when $A$ and $B$ are present at their initial amounts, say $A_0$ and $B_0$, to when the reaction is 99% complete.  

When $[\ce{A}]\gg[\ce{B}]$, the reaction will be 99% complete when 99% of $B$ is gone, i.e. when $B = 0.01B_0$.  

At that point, $$\frac{dx}{dt} = k[A][B]^2 = k(A_0 - \frac{0.99}{2}B_0)(0.01B_0)^2 $$

Now since $A\gg B$ also means $A_0\gg B_0$, then it follows that $(A_0 - \frac{0.99}{2}B_0) \approx A_0$.  Thus, when the reaction is 99% complete and $B$ is limiting,

$$\frac{dx}{dt} \approx kA_0(0.01B_0)^2 = kA_0 (0.0001)B_0^2$$

If we lump $k$ and $A_0$ into a single parameter, $k_{app}$, then we can write $\frac{dx}{dt} = k_{app}B^2$, which shows that when $B$ is limiting, the reaction is 2nd order. 

The case for $A$ being limiting shows the reverse.