I have a question about the physical interpretation of a particular limiting case in a problem. Please note my question is not about how to actually solve the problem, but rather an entirely conceptual question.
Consider a block on a wedge inclined at angle $\theta$. The coefficient of friction between the block and plane is $\mu$. What is the maximum acceleration, $A$, that we can accelerate the wedge at so that the block remains on the wedge without sliding?
At large accelerations, we can expect the friction on the block to point down the slope. From this, I was able to derive that the maximum possible acceleration has magnitude $$A=\frac{\sin{\theta}+\mu\cos{\theta}}{\cos{\theta}-\mu\sin{\theta}}g.$$ Here is my question. For the magnitude above to be positive, we require $\cos{\theta}-\mu\sin{\theta}>0$, or $\tan{\theta}<\frac{1}{\mu}$.
But what if we are given that $\theta$ is large enough so that $\tan{\theta}>\mu$? Combining the two conditions, we must have $$\mu<\tan{\theta}<\frac{1}{\mu}.$$ In particular, we have that $\mu<\frac{1}{\mu}$, which is only true if $\mu<1$ for positive $\mu$.
This means that if $1<\mu<\tan{\theta}$, the magnitude of the maximum acceleration is negative, which doesn't make any sense. It seems that such a maximum acceleration does not exist in this case, and I can make the acceleration arbitrarily large without the block moving. How can I physically interpret this?