What happens when an expression for a physical quantity becomes undefined? Now I am taking a quite a specific example here. Let's consider a circular race track banked at an angle of $\theta$. The coefficient of friction between the track and the tire is $\mu$. What's the maximum speed that the car can achieve without skidding out?
Now to solve this problem we draw the car's free body diagram, consider centripetal acceleration and after a lot of math we come up with the expression of $$v_{\rm max} = \sqrt\frac{g\,r\,(\tan \theta + \mu)}{(1-\mu\,\tan\theta)}$$
Now the my question is that what happens when $$\mu\,\tan \theta > 1$$ Obviously, the expression for $v_{\rm max}$ becomes undefined but I would like to know that in physical world, what would be the fallout of this?
 A: The key part is "max speed without skidding out". We know intuitively that if the speed increases, the car will skid out, moving radially outwards (up the bank).
When you solved for that expression, you placed in the condition that the all forces that move the car upwards cancel with all the ones that move it downward (upwards and downwards defined depending on coordinate system).
So answering it mathematically, a good way to think about this is to see what happens when we vary $\theta$, as this is the free parameter. I do just want to correct one thing you said: the expression for $v_{max}$ actually becomes undefined for $\mu\;tan\,\theta \geq 1 $.
Anyway, Let's say $\mu = 0.5$. Start with $\theta = 0$ and increase the $\theta$ gradually, which corresponds increasing the incline. As you do, the max speed you can go before skidding out increases. This makes sense, a car would struggle to move up the incline. The speed required to skid out increases asymptotically with $\theta$ and at some incline, $\theta=63.43^\circ, \mu\;tan\,\theta=1$ and there is no finite speed high enough to skid out.
